Calculus 2, part 2 of 2: Sequences and series

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.

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 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.

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

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.

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.

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.

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.

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.

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).

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.

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.

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 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.

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.

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 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.

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.

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

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

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.

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.

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.

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.

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.

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

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 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.

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

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

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.

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

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 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.

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.

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 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 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.

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.