Calculus 2, part 2 of 2: Sequences and series

Explore a recap of sequences from calculus one part one, covering limits, boundedness, monotonicity, and convergence, with the squeeze theorem and Weierstrass principles applied to infinite limits and indeterminate forms.

Explore why sequences matter: they sharpen function analysis, enable discretization from continuous to discrete models, support approximations, and underpin two dimensional interdependent sequences and linear discrete dynamical systems.

Revisit arithmetic and geometric progressions, define constant difference and ratio, and present the nth-term formulas a_n = a + (n-1)d and a r^{n-1}, linking to linear and exponential behavior.

Master arithmetic and geometric sums by deriving Gauss’s partial-sum formula and computing geometric sums for sequences with ratio q, preparing for series in later coursework.

Explore continuous versus discrete models, showing how linear ordinary differential equations with constant coefficients and linear recurrences share a characteristic equation, yielding exponential and geometric solutions.

Calculus 2, part 2, explores old and new methods for convergence of sequences, from epsilon definitions and squeeze theorems to conjugates and ratio tests, with a preview of Riemann-sums interpretation.

Apply rules of limits and real arithmetic to evaluate linear combinations of a_n and b_n, with a_n -> 1 and b_n -> -1, including cases yielding zero or plus infinity.

The lecture tackles indeterminate forms and standard limits, solving exercise 2 by identifying the fastest growing term as n^5 and dividing through, yielding a limit of zero.

Resolve the indeterminate form infinity minus infinity by using conjugates and the difference of squares, multiplying by the conjugate, canceling terms, and dividing by n to obtain two.

Apply the conjugate and the difference of two squares to compute the limit in exercise four, then divide by sqrt(n) to obtain one half.

Determine the base limit with standard limits at infinity, which yields 1/2; apply the extended-reals power rule to conclude that (base between 0 and 1) raised to infinity equals zero.

Use the squeeze theorem to compute limits of x_n, y_n, and z_n, compare methods, and apply Riemann sums to evaluate a_n, explaining why it fails for z_n.

Apply the squeeze theorem to the nth roots of the sum of the nth powers of a1 through am to show the limit equals A, the maximum of the numbers.

Prove theorem: if a_n tends to zero with a_n not zero and a_n greater than minus one, then limit as n → ∞ of (1 + a_n)^{1/a_n} equals e.

Explore indeterminate forms of one to the infinity in various settings, and apply the nth roots theorem for positive convergent sequences to evaluate limits.

Explore how raising one plus a_n to the reciprocal power 1/a_n yields e as a_n tends to zero, using a_n = -3/n to obtain e^-3.

Explore solving indeterminate forms of the type one to infinity for sequence limits, using extraction and the theorem to show the limit equals e^{-2}.

Resolve an indeterminate form by rewriting the base as 1 plus a small term and applying a limit theorem to the sequence, showing the limit is e squared.

Solve a difficult limit in sequences and series by using a clever factorization trick and extended-real arithmetic, showing that (1+1/n)^n tends to e and handling infinity and indeterminate forms.

Explore the Weierstrass monotone convergence theorem for increasing and decreasing bounded sequences, review boundedness, supremum and infimum, and practice with a warm-up exercise.

Explore proving (eventual) monotonicity by analyzing quotients for positive sequences, comparing differences and quotients, with Weierstrass theorem and recurrence methods guide when to use each approach.

Analyze quotients to prove eventual monotonicity of the positive sequence a_n = 2^n/n^n, show it is decreasing and bounded, apply Weierstrass theorem to deduce a zero limit.

Demonstrates that the subsequence x_n = (1+1/(2n))^{2n} is increasing, using subsequence concepts and the monotonicity of power functions, for a sequence that tends to e.

The lecture demonstrates eventual monotonicity using a bit of Weierstrass cheating. It derives L by solving L = sqrt(15 + 2L), proves boundedness and increasing, and converges to 5.

Determine the limit of the nested roots sequence x1 = sqrt(c), x_{n+1} = sqrt(c + x_n) by establishing boundedness and monotone increase, then solve l = sqrt(c + l) for the positive root.

Apply the monotone convergence theorem to the recursively defined sequence a_{n+1} = a_n/2 + 1/(2 a_n); show it is decreasing and bounded below by 1, and obtain the limit equals 1.

Prove that the recursively defined sequence is eventually decreasing using the quotient test, confirming positivity and deriving a threshold n > 13/3 (n ≥ 5) for decrease.

Analyze a recursively defined sequence where a_{n+1} = a_n times a factor that vanishes at n = 5, making a6 = 0. Conclude the sequence is eventually constant and monotone.

Apply Riemann integrals to show the harmonic sum sequence SN = 1 + 1/2 + ... + 1/n is increasing and unbounded, so the harmonic series diverges to infinity.

The lecture generalizes the Weierstrass theorem to unbounded monotone sequences, showing that increasing sequences diverge to plus infinity and decreasing sequences diverge to minus infinity in the extended real line.

Compare BN, the partial sums of a_n = 1/√n, with SN, the partial sums of 1/n, using a squeeze-type argument to show BN diverges to infinity. BN is increasing.

Demonstrate that the sequence a_n is increasing and bounded above by two, and, via a telescoping cancellation, determine its limit as two.

In problem 8, compare bn with a_n to show bn is bounded above by 2 and increasing. The inequality bn ≤ a_{n-1} gives a limit exists between 0 and 2.

Examine the product form of a sequence in problem 9, proving a_n is strictly increasing and unbounded, with factors just above one yielding a_n = n+1 and tending to infinity.

Show that the sequence a_n is decreasing and bounded below by zero. Use the Weierstrass theorem to conclude the limit exists, while the relation cannot determine its value.

Demonstrate that BN is increasing by adding positive terms, and bound BN by three with a geometric sum q = 1/2 to show convergence to e.

Analyze how the difference method reveals monotonicity in geometric progressions, contrast with the quotient method’s limits for negative first elements, and outline case analysis by a and q.

Shift to a functional approach to sequences and apply derivatives and L'Hôpital's rule to analyze monotonicity, limits, and geometric progressions.

Apply differential calculus to sequences by turning them into a corresponding function, analyze monotonicity and limits with derivatives and L'Hôpital's rule, and determine boundedness and convergence for the given exercise.

Explore how using a function approach versus direct sequence methods affects monotonicity and limits, through four examples with differential calculus and L'Hôpital's rule, illustrating when each method excels.

Explore limits of sequences using L'Hôpital's rule and logarithmic techniques, analyze monotonicity, and verify with plots, linking calculus concepts to sequences and series.

We show that the limit of the logarithm diverges to infinity because a geometric progression grows without bound while cosine stays bounded, and the log is continuous.

Use the logarithm quotient rule to rewrite the difference in the sequence's terms as log((3n^2-3)/(5n^2+5)); divide by n^2 to get a 3/5 limit and apply continuity to obtain log(3/5).

Explore the nth root of n as a sequence, prove n^(1/n) is eventually decreasing, and use function analysis of f(x)=x^(1/x), limits, and key comparisons like e^pi vs pi^e.

Learn how functions illuminate sequences and series using limits and differential calculus. See partial sums, the integral test, and area under 1/x with a_n=1/n as a spoiler.

Explore the Stoltz-Cesaro theorem: formulation, proof sketch, and its role as a discrete analogue of L'Hôpital's rule for sequences with infinity limits.

Apply the Stolz–Cesàro theorem to x_n = a^n and y_n = n with a>1, showing (x_n−x_{n−1})/(y_n−y_{n−1}) tends to infinity, hence a^n/n tends to infinity.

Apply Stoltz-Cesaro to prove Cauchy's theorem: if a_n converges, its arithmetic means converge to the same limit. Then show a divergent a_n with convergent means, proving the converse false.

Prove a corollary of Cauchy's theorem: if the differences a_n - a_{n-1} converge to g, then a_n/n converges to g. Provide counterexample a_n = (-1)^n showing the converse can fail.

Apply the corollary from V53 to show log n over n tends to zero by examining log(n+1) - log n and using continuity, contrasting with L'Hôpital and Cesàro proofs.

Apply the Stoltz-Cesaro theorem to the natural-number p case to obtain the limit 1/(p+1) by analyzing the quotient of differences x_n and y_n; connects to the earlier Riemann-sums approach.

Explore using formulas or riemann sums to evaluate the limit for p = 1, 2, 3, yielding 1/2, 1/3, 1/4, and contrast with cesaro to show riemann sums generality.

For a positive convergent sequence a_n, the geometric means g_n converge to g; the converse fails, as shown by a counterexample where g_n converges but a_n does not.

Establish a corollary: if a_n are positive and a_{n+1}/a_n tends to g, then the nth roots of a_n tend to g, using geometric means of the quotient sequence and cancellations.

Apply the corollary from video 58 to positive sequences by using quotients of consecutive terms; define x_n = n^n/n!, x_{n+1}/x_n → e, and nth roots of x_n tend to e.

Explore the ratio test for sequences, where the limit of |a_{n+1}/a_n| equals g. If g<1, a_n tends to zero; if g>1, it diverges to infinity; if g=1, test is inconclusive.

Apply the ratio test to sequences by examining the quotients a_{n+1}/a_n; for a_n = n! / a^n with a=10, the quotients tend to infinity, so a_n diverges to plus infinity.

Apply the ratio test to sequences by examining a_{n+1}/a_n for the problem with a_n = (n+2)!/(2n+1)!, showing the limit is zero and establishing convergence.

Explore three methods to solve the ratio test for sequences in problem 6: Weierstrass theorem, a_{n+1}/a_n via ratio test, and a squeeze theorem approach. All show a_n tends to zero.

Explore why the ratio test's case three, where g equals one, is inconclusive, and see four examples showing convergent and divergent sequences despite g = 1.

Explain why the ratio test for sequences is not an if-and-only-if statement; examine two constructed sequences showing a_n -> 0 but the quotients a_{n+1}/a_n can diverge, illustrating the test's limits.

Learn to verify a linear recurrence of order two. Verify the explicit form x_n = 2^n + 1 for x_{n+1} = 2x_n minus 1 with x_1 = 3.

Explore solving a first-order linear non-homogeneous recurrence by iteration, deriving the explicit formula x_n = 4 + n(n+1)/2 for x_n with x_0 = 4.

Apply telescoping techniques to a first-order recurrence to obtain a closed form solution; rewrite as differences, cancel intermediate terms, yielding x_n = 4 + n(n+1)/2.

Explore linear recursion of order two with two initial terms, solving for x_n as geometric progressions: 2^n, 3^n, or 2^n+3^n.

Explore linear recursion of order two and linearity; show geometric progressions solve the recurrence when the base is a root of the characteristic equation, and combinations of solutions remain solutions.

Derive the closed formula for linear recursions of order two with real roots, expressing f_n as a linear combo of two geometric progressions and determining constants from f0 and f1.

Derive the closed form for a linear recurrence of order two by solving the characteristic equation, yielding x_n = c1 3^n + c2 (-2)^n, a family of solutions without initial conditions.

Solve a second-order linear recurrence with x0=2 and x1=3 by using the characteristic roots 5 and 2; derive x_n = (-1/3) 5^n + (7/3) 2^n.

Solve a linear recursion of order two with x0=1 and x1=2, deriving roots 1±√2 and a closed form yielding natural numbers.

Derives the Fibonacci closed form via the characteristic equation, confirms f0=0, f1=1, solves for c1=1/√5 and c2=-1/√5, and notes indexing from zero.

Explore practical applications of sequences, including numerical methods and recurrence relations, with examples from calculus and linear algebra, using matrices, eigenvalues, and eigenvectors.

Explore how to model real problems using Fibonacci sequences and recursive sequences, culminating in the recurrence x_n = x_{n-1} + x_{n-2} with x_1 = 2 and x_2 = 3.

Explore change factor and compound interest using precalculus formulas, including geometric sums and annuities, with insights into continuously compounded interest and the exponential relationship.

Model a lake’s fish population with a recursive sequence. Use P_n = 1.06 P_{n-1} - 150 and P_0 = 2000, applying a geometric sum approach to compute P_12.

Show that infinite decimal expansions form a convergent sequence between 0 and 1, and use a geometric sum with Weierstrass theorem to justify nine repeating equals one and pi digits.

Applies Newton's method to approximate the square root of two via a_{n+1}=a_n/2+1/a_n with a0=2, proving the sequence stays above sqrt(2) and converges to sqrt(2).

Explore a sequence defined by a_{n+1} = (a_n + 1)/2 on [0,1], shown as the midpoint of [a_n,1], proving it is increasing and bounded above by 1 with limit 1.

Explore how the recursive sequence s_{n+1} = sqrt(2 + s_n) with s0 = 1 models perimeters of regular polygons inscribed in the unit circle and yields a pi approximation.

Study sequences defined with the help of Riemann integrals and apply a geometric progression bound, monotonicity of integration, and a squeeze argument to determine their limits.

Show that a_n equals the integral from 0 to 1 of (x^2+1)^n dx, and diverges to infinity by a lower bound on [1/2,1], since f is increasing and f(1/2)=5/4, giving a_n ≥ (1/2)(5/4)^n.

Analyze sequences defined by Riemann integrals that tend to zero. Bound a_n by the maximum of x(1−x) on [0,1], use monotonicity of t^n to show a_n ≤ (1/4)^n and thus a_n→0.

The lecture shows that a sequence defined by a Riemann integral tends to zero, using an epsilon proof and a two-part integral split based on sine monotonicity.

Apply the ratio test to the quotient a_{n+1}/a_n, showing its limit is 2/e < 1, hence a_n converges to zero; verify via the Weierstrass theorem.

Explore a clever trick to prove the limit of (1+n+n^2)^{1/n} equals one, using binomial expansion and the squeeze theorem to bound x_n.

Use properties of logarithms and integral bounds to show log(n!) / (n log n) tends to 1 by the squeeze theorem.

Discover how the difference between consecutive squares equals an odd number. Also learn that the sum of odd numbers from 1 to 2n-1 equals n^2, illustrated with unit squares.

Analyze subsequences of convergent sequences and use the geometric sum to evaluate a limit; show that the sequence tends to two.

Apply a trigonometry trick to evaluate a product limit using the sine double-angle formula and sin x / x → 1, showing the limit equals sin(phi)/phi for phi ≠ 0.

demonstrates the divergence of the sequences x_n = cos n, y_n = sin n, and z_n = tan n by limit-based contradiction, using recursive relations and the Pythagorean identity.

Two remarks to video 94 offer a simpler solution to the alpha and beta system for x_n, y_n, z_n, showing that x_n and y_n do not converge.

Explores real number sequences and sequences in metric spaces, examining monotonicity, boundedness, convergence, subsequences, limit superior and inferior, Bolzano-Weierstrass, and Cauchy property.

Explore how boundedness, convergence, and Cauchy sequences behave in metric spaces, and distinguish which sequence properties generalize beyond real numbers.

Bound sequences in metric spaces by requiring all terms lie within an open ball centered at y0 with radius m, while monotonicity requires an order that metric spaces may lack.

Explore convergence of sequences in real numbers and metric spaces, using distances and open balls, and formalize subsequences by index sequences and order.

Present the Cauchy property and fundamental sequences using distance and epsilon, showing that for large indices elements get arbitrarily close, and that convergence depends on the ambient space.

Compare real-number sequences with sequences in metric spaces, highlighting boundedness, limits, and the Cauchy property, while noting the limits of order-based theorems and subsequences in general metric spaces.

Explore a roadmap linking boundedness, monotonicity, convergence, the Cauchy property, and subsequences with theorems that reveal how these features relate in real numbers and metric spaces.

This lecture proves that every convergent sequence is bounded and its subsequences converge to the same limit, and that monotone and bounded real sequences converge to their infimum or supremum.

Explore the Cauchy property and convergence in metric spaces, and show that convergent sequences are Cauchy. Demonstrate fundamental sequences and completeness with pi and sqrt(2) approximations, highlighting R versus Q.

Explore how fundamental sequences are bounded in metric spaces, show that convergence implies boundedness but not vice versa, and examine the Cauchy property and related examples.

Explore how the Cauchy property and fundamental sequences interact, showing proximity of consecutive terms is not enough, via a_n = sqrt(n) where differences tend to zero but a_n is unbounded.

Prove that every sequence of real numbers contains a monotone subsequence, using completeness and a two-case approach to yield either an increasing or a non-increasing subsequence.

Bolzano-Weierstrass' theorem states that every bounded sequence of real numbers has a convergent subsequence. Its proof combines a monotone subsequence theorem and the Weierstrass theorem to show convergence.

Prove that every fundamental sequence of real numbers converges in R, using Bolzano–Weierstrass, boundedness, and the triangle inequality to extend convergence from a subsequence to the whole sequence.

Learn a new construction of the real numbers using equivalence classes of Cauchy sequences of rational numbers, where real numbers are limits of convergent sequences under a zero-difference relation.

Explore accumulation points as limits of subsequences in real sequences, including when bounded sequences with one accumulation point converge, and why a single accumulation point does not guarantee convergence.

Prove that every bounded real sequence with exactly one accumulation point is convergent, using contrapositive and Bolzano-Weierstrass, and analyze subsequences and accumulation points.

Apply a new method to compute the old limit of kth roots: show convergence of a_n^(1/k) to a^(1/k) using boundedness, accumulation points, Bolzano-Weierstrass, and non-negativity for even k.

Discover limit superior and limit inferior as the supremum and infimum of a sequence's accumulation points, and relate them to tails, convergence, and reading notes.

Apply the nested intervals theorem to closed intervals with diameters tending to zero, yielding a unique intersection point; left endpoints converge to the supremum and right endpoints to the infimum.

Apply the nested intervals theorem to the alternating partial sums a_n, showing convergence and bounding the limit l between a_{2k} and a_{2k+1} with decreasing interval length 1/(2k+1).

Compute the exact limit of the sequence from the previous lecture, showing it converges to the logarithm of two, and apply an error estimation using Riemann integrals.

Explore complete metric spaces and Cauchy sequences, and see how convergence and continuity translate to metric spaces for the upcoming real analysis course.

Define convergent and divergent series via partial sums and limit. Learn when the limit exists or diverges, and study geometric, p, harmonic, alternating, and telescoping series with the integral test.

Explore three simple examples of convergent, divergent, and alternating series. Learn how Cesàro sums assign a value via averages of partial sums and why associativity can mislead, as with Grandi's paradox.

Revisit classic series from sequences, including harmonic, telescoping, and alternating series, using partial sums to illustrate convergence, divergence to infinity, and exact values.

Move from a sequence to its series by forming partial sums. Compute the first four partial sums from the general term and see the sum as the limit.

Derive the general term a_n from a given series using sigma notation and index choices, and study examples of reciprocal odd and even terms, telescoping and alternating series.

Explain that if a series converges, its terms tend to zero, but this is not a sufficient condition. Use contrapositive to test divergence and compare with improper integrals.

Apply the necessary condition for convergence to show that the series ∑ n/(n+2) diverges since a_n does not tend to zero; note that meeting the condition is not sufficient.

Starting from different indices changes the sum of a series, unlike sequences where the limit is unaffected by initial terms; convergence remains unaffected by removing early terms.

Learn the if and only if condition for convergence of infinite series by examining heads, tails, and remainders, and see how the remainder convergence characterizes the original sum.

The Cauchy condition for a series states that it converges iff, for each epsilon, there exists n_epsilon with |s_{n+k}-s_n| < epsilon for all n ≥ n_epsilon and k ≥ 1.

Explore arithmetic series and convergence: only the trivial progression with a=0 and d=0 converges; all nonzero d diverge to plus or minus infinity; geometric series will be more interesting.

Explore geometric series and geometric progressions, deriving the partial sum formula and conditions for convergence: |q|<1; discuss special cases a=0, q=0, q=1 and divergent behavior for |q|>=1.

Explore geometric series, learn to identify the common ratio q and first term a, and apply factoring or remainder methods to sum series with different starting indices.

See a geometrical illustration of summability for a geometric series using a unit square, showing how 1/2^n sums to 1 and how omitting the first term changes the total.

Explore how to add, scale, and take differences of series elementwise, using partial sums and limit laws. Note that products of series are not simple.

Apply the linear-combination theorem to express a series as a sum or difference of geometric progressions, determine convergence from quotients between -1 and 1, and compute sums 5/6 and 26/15.

This lecture proves p-series converge when p>1 and diverge when p≤1, using comparison to the harmonic series, monotone partial sums, and geometric bounds.

Explore alternating series, learn the Leibniz criterion for convergence with non-negative, non-increasing a_n and limit zero, and apply to classic examples like the alternating harmonic series.

Apply Leibniz criterion to convergent alternating series to estimate sums via even and odd partial sums. Using the alternating harmonic series, approximate to one decimal place with S9–S12, yielding 0.7.

Explore telescoping series and their convergence by transforming terms into differences and canceling adjacent terms. Analyze partial sums and limits to determine when the series converges and its sum.

Apply a telescoping trick by rewriting ones as differences, split fractions to reveal cancellations, and compute the partial sums that converge to 1/2.

Explore telescoping series and their convergence through example 2, using a trick and a more standard method, and apply a linear-combination theorem to find the sum as one quarter.

Use partial fraction decomposition to evaluate a telescoping series and determine its convergence, showing that the sum equals one fourth.

Explore alternative proofs that the harmonic series diverges to plus infinity using subsequences of partial sums and Weierstrass theorems, and show the p-series with p=1/2 diverges without harmonic comparison.

Study a nice alternating series that diverges to plus infinity, using pairwise subtraction and a geometric interpretation tied to the harmonic series.

Turning sequences into infinite series by using partial sums and a telescoping approach; form a geometric series with ratio -1/2 to obtain the limit 2/3.

Turn sequences into infinite series through a geometric introduction to Euler's constant, defining gamma as the convergent number between 0 and 1 from partial sums of the harmonic series.

Explore absolute and conditional convergence using associated positive and negative term series and the Cauchy criterion. Apply these ideas to harmonic and alternating harmonic series, and to geometric series.

Explore arithmetico-geometric series, derive their partial sums, and determine convergence for |x|<1, revealing the sum equals 1/(1-x)^2 through two methods.

Examines arithmetico-geometric series, derives a closed form for partial sums s_n, uses difference of squares and geometric sums, and shows convergence occurs for |x|<1 with the sum equal to 2/(1−x)^3.

Explore arithmetico-geometric series and derive a closed formula for partial sums, even when the series diverges to plus infinity, using Gauss's trick and a x=2 substitution to obtain S_n.

Compute the series n/2^n using methods. Split into a geometric and convergent part to get 2; or factor out one half and apply a formula with x=1/2 to get 2.

Analyze the arithmetico-geometric series in problem 5 by decomposing terms into geometric and scaled series, assuming convergence, and apply the relevant theorem to derive the sum equals six.

Explore the convergence of series in calculus ii through tests such as geometric, p, ratio, direct and limit comparison, absolute and alternating criteria, with practical exercises and a clear workflow.

Use the Leibniz alternating-series criterion to test three series: the first and third converge as a_n decreases to zero, while the second diverges since a_n tends to one third.

Assess the alternating series with a_n = ln n / n, showing a_n tends to 0 and decreases from n ≥ 3, yielding conditional convergence by Leibniz criterion.

Identify this as a geometric series with ratio 1/1000 and a = 5. The series converges and its sum is 5/(1-1/1000) = 5000/999.

Identify whether four p-series converge or diverge, classify alternating cases, apply the p-series test and Leibniz criterion, and distinguish absolute from conditional convergence.

Learn the direct comparison test for positive-term series, prove and apply it using upper bounds for convergence and lower bounds for divergence, with p-series and geometric series as guides.

Apply the comparison test to determine convergence of a series by bounding it above with a convergent p-series 1/n^2, and recall p-series and geometric series criteria.

Apply the comparison test to prove divergence of the series sum from n=1 to infinity of (3n+5)/(2n^2+1) by bounding below by 1/n, a divergent harmonic (p-series).

Apply the comparison test to a positive-term series by bounding its terms above with a convergent geometric series (1/2^n), showing the original series converges.

Apply the comparison test to the series log n / n, bound below by the harmonic series from n = 3, and conclude divergence to infinity.

Apply the comparison test to exercise 9 in sequences and series, starting at n = 2, using log n to bound terms and compare with harmonic series to show divergence.

Apply the comparison test to the series sin(n)/n^2 by analyzing the absolute values. Show that |sin n|/n^2 is bounded by 1/n^2, so the series converges absolutely.

Use the comparison test on the series 1/√(n^2-3) and bound below by 1/n, yielding divergence via the harmonic series.

Apply the limit comparison test to positive series to determine convergence or divergence by comparing to known p-series or geometric series, and learn the role of the limit L.

Apply the limit comparison test to the series with a_n = (n+5)/(n^3-2n+3) and b_n = 1/n^2; the limit equals one, so the series converges.

Use the limit comparison test to compare a_n = n/(n^4−2) with b_n = 1/n^3, showing convergence by a p-series with p=3—initial negative terms do not affect convergence.

Apply the limit comparison test to exercise 15, showing a_n behaves like a convergent p-series with p = 3/2. Conclude convergence by establishing limit of a_n over b_n equals 1.

Apply the limit comparison test to exercise 16, comparing the series with 1/(3n) and concluding the original diverges as a scaled harmonic series.

Apply the limit comparison test to exercise 17, using the fact that the nth root of n tends to one to compare to the harmonic series and conclude divergence.

This example demonstrates that the direct comparison test confirms convergence for a_n, where a_n = (1+sin n)/n^2. The limit comparison test fails here because the limit does not exist.

Explore the comparison test two lemma for series with positive elements: if a_{n+1}/a_n <= b_{n+1}/b_n and b_n converges, then a_n converges; if a_n diverges, b_n diverges.

Explore d'Alembert criterion for series with positive elements. Converges when quotients stay below q between zero and one for almost all n; diverges when quotients reach one for almost all n.

Explore the d'Alembert criterion with limit superior and limit inferior, and see how accumulation points determine convergence, divergence, or an inconclusive result.

Apply the ratio test to positive sequences by examining d = lim a_{n+1}/a_n; if d<1 converges, if d>1 diverges, and if d=1 inconclusive (via limit superior and limit inferior).

Apply the ratio test in calculus 2, part 2, exercise 19 to a_n = n/2^n; show d_n tends to 1/2, thus the series converges.

Use the ratio test on the series with a_n = n^2/2^n to show convergence. Compute a_{n+1}/a_n, which tends to 1/2, hence the series converges.

Explore the ratio test on the series sum x^n / n!, proving absolute and regular convergence for all x, and relate to the Maclaurin expansion of e^x.

Examine why the ratio test fails for p-series with a_n = 1/n^p, since a_{n+1}/a_n → 1, so the test cannot decide convergence, unlike the integral test.

Apply the ratio test to the series a_n = n! / n^n, cancel terms, and show d = a_{n+1}/a_n → 1/e; since 1/e < 1, conclude convergence.

Apply the ratio test to the series with a_n = 2 n!/(n!)^2, revealing cancellations and a ratio limit of 4. Conclude the series diverges.

Apply the ratio test to exercise 25's series, simplify a_{n+1}/a_n to reveal the limit d = 3/e, which is greater than 1, and conclude the series diverges.

Apply the ratio test to exercise 26 for a series with a_n expressed via factorials and odd-number products. Show that a_{n+1}/a_n = (n+1)/(2n+1) → 1/2, so the series converges.

Apply the ratio test to the series with a_n = 1/(n 2^n). Compute the limit a_{n+1}/a_n, get d = 1/2, and conclude convergence, comparing with a geometric series.

Apply the ratio test to the given series, compute a_{n+1}/a_n, and find the limit is 1/4. Conclude the series converges.

Apply the ratio test to exercise 29, determine a_n and a_{n+1}, and conclude the series diverges using the limit comparison and direct comparison tests.

Apply the ratio test to exercise 30, noting the inconclusive limit of 1, then conclude the series diverges to plus infinity by comparing to numbers approaching e from below.

Explain Cauchy criterion for series with positive elements using c_n as root of a_n, showing convergence when c_n <= q < 1 and divergence if c_n >= 1 infinitely often.

Use the Cauchy criterion using limit superior to decide convergence. If limsup CN < 1, the series converges; if > 1, it diverges; if equal to 1, it's inconclusive.

Apply the root test to positive sequences by analyzing nth roots. Conclude convergence if the limit is below one; conclude divergence if above one; otherwise inconclusive.

Apply the root test to exercise 31's sequence a_n = (n/(n+1))^n, compute its nth root to obtain limit c = 1/e, and conclude the series converges (not the sum).

Apply the root test to exercise 32, where a_n is a fixed positive base to the nth power. Since c_n tends to zero and is below one, the series converges.

Apply root test to series in exercise 33, use nth roots to show the limit tends to a e divided by three, and conclude convergence since it’s less than one.

Apply the root test to exercise 34, using a supremum form and odd/even subsequences to find two limit points, show the limit superior is less than 1, hence convergence.

Explore the integral test for positive-term series, linking convergence to the improper integral from one to infinity, and compare visual proofs with formal writing, using function sequence connections.

Apply the integral test to p-series to determine convergence: they diverge for p ≤ 1 and converge for p > 1, via improper p-integrals and comparison.

The lecture contrasts integral test with the comparison test and limit comparison test for a p-series, showing divergence by comparing to 1/√n and guiding when to choose the optimal test.

Apply the integral test to a_n = 1/(n log n) starting at n=2. The improper integral diverges, so the series does too; generalize to 1/(n (log n)^p) with p>1.

Apply the integral test to show convergence of the series 1/(n^2+1) and bound its sum by pi/2, using the arctangent integral.

Apply the integral test and the limit comparison test to the series with a_n = n/(n^2+1). Compare to the harmonic series and evaluate ∫_1^∞ x/(x^2+1) dx to show divergence.

Apply the integral test to a known telescoping series with sum equal to one, using f(x)=1/(x(x+1)) on [1, ∞) to obtain the improper integral value ln 2 and confirm convergence.

Apply the integral test to exercise 42 and use the fact that nth roots of a positive constant tend to one; compare to a p-series with p=2 to show convergence.

Apply the integral test to exercise 43 for the series with f(x)=1/(x log x (log log x)^2). Show it's decreasing and use t=log x, u=log t substitutions to prove convergence.

Start with direct or limit comparison, then ratio or root tests; use the integral test when needed. Study Dirichlet and Abel’s tests and practice with Schaum’s outlines.

Apply the conjugate method to simplify the general term, verify positivity, and use a comparison test against a p-series with p = 3/2 to establish convergence.

Apply the ratio test to the series a_n = n^2 sin(pi/2^n). The limit a_{n+1}/a_n equals 1/2, so the series converges.

Explore convergence of two series by applying limit comparison test, ratio test, root test, and integral test, illustrating when these tests yield convergence, divergence, or inconclusive results.

Use the comparison test to prove the series converges to a number between 0 and 1, linking gamma to e via the monotonic behavior of a_n and c_n.

Investigate the alternating harmonic series and prove its sum equals log two by analyzing partial sums, the subsequence a_{2n}, and the gamma connection, using convergence results.

Show how a sequence tends to zero by proving the related series converges using the root and ratio tests. Conclude the limit from the necessary condition for convergence.