Introduction to Calculus 3: Infinite Sequences and Series

Explore infinite sequences and infinite series, including convergent and divergent sequences, and learn properties such as increasing, decreasing, and bounded as you define and analyze sequences.

Define a sequence and its nth term a_n, and illustrate infinite sequences with examples like n/(n+1), sqrt(n-3), the Fibonacci sequence, and (-1)^n, plus convergence via limits.

Explore the convergence of a sequence by defining its limit as n approaches infinity, distinguishing convergent and divergent cases, and applying limit rules, the squeeze theorem, and sequence-to-function connections.

Explains eight sequence examples, determining convergence or divergence and limits, using factoring, the squeeze theorem, L'Hôpital's rule, and growth rate comparisons.

Explore monotonic and bounded sequences, prove convergence using the monotonic sequence theorem, and examine examples including a decreasing sequence and a recurrence-defined sequence that converges to six.

Explore section three by introducing infinite series, geometric and p-series, and applying the seven tests—divergence, integral, direct comparison, limit comparison, alternating, ratio, and root—to assess convergence or divergence.

Define a series as the sum of an infinite sequence using sigma notation and partial sums; recognize convergence via the limit of partial sums in examples like pi/4's alternating series.

Explore geometric series, identify a nonzero initial term a and a common ratio r, determine convergence when |r| < 1, and derive the sum a/(1 - r) with examples.

Master telescoping series by canceling terms in pairs, decompose using partial fractions or log properties to obtain nth partial sums, and decide convergence or divergence, with harmonic series next.

Explore the harmonic series, an infinite sum of unit fractions, proving its divergence by regrouping terms and linking to convergence tests for series.

Explore the divergence test for infinite series and learn that a nonzero or nonexistent limit of a_n guarantees divergence, while a zero limit remains inconclusive.

Use the integral test to link series to the improper integral of a continuous positive decreasing function; verify convergence for 1/(n^2+1) and divergence for (ln n)/n, then summarize p-series.

Apply the integral test to estimate the sum of a convergent series. Bound the remainder between the improper integrals and use partial sums to determine the number of terms needed.

Apply the direct comparison test to positive terms. If a_n <= b_n and b_n converges, then a_n converges, and if a_n >= b_n and b_n diverges, then a_n diverges.

The alternating test determines convergence for alternating series by requiring decreasing absolute terms and a limit of zero, illustrated with convergence, divergence, and a derivative-based monotonicity check.

Apply the alternating series estimation theorem to bound the remainder by the next term, and use it on the series sum (-1)^n/n! for a three-decimal estimate.

Explore absolute and conditional convergence of alternating series using absolute value criteria, p-series, and the comparison test, with examples like sum (-1)^{n-1}/n^2 and cosine n over n^2.

explains the ratio test for series by evaluating the limit of |a_{n+1}/a_n| and its three outcomes—convergent, divergent, inconclusive—using factorials and exponentials as examples.

Apply the root test to series with terms like (2^n+3)/(3^n+2); the limit of the nth root of |a_n| equals 2/3, so the series converges absolutely.

Summarize the seven convergent tests for series, including the integral, direct comparison, limit comparison, alternating, ratio, and root tests. Explore geometric and p-series, and telescoping series with partial fractions.

Explore power series as functions of x, learn convergence tests and the interval of convergence, and discover three methods to derive a function’s power series representation.

Explore the power series definition, centered around a, with two examples. Use the ratio test to determine the domain and radius of convergence and interval of convergence.

Examine two examples of radius and interval of convergence using the ratio test, and determine where the series converge or diverge along endpoints.

Represent functions as power series by rewriting them as geometric series and differentiating or integrating term by term, determining convergence, with examples such as 1/(1−x), 1/(1+x), and x^3/(x+1).

Explore how to represent functions as power series through Taylor and Maclaurin expansions, derive coefficients via nth derivatives, and determine radii of convergence using the ratio test.

Verify when a function has a power series representation by analyzing the remainder in Taylor series; apply Taylor's inequality to ensure convergence and derive the Maclaurin expansion of e^x.

Learn to express functions as power series and prove their Taylor expansions via direct computation. Include remainder analysis, radius of convergence, and examples like sine x and binomial series.

This lecture teaches method two for Taylor expansions—differentiating or integrating power series term-by-term—with examples on cosine, x cos x, arctangent x, and inverse hyperbolic sine approximation.

Review two methods for obtaining Taylor and Maclaurin series, then memorize key Maclaurin series such as e^x, sin x, cos x, arctan x, ln(1+x), and the binomial series.

Use summation, multiplication, and division of power series to build Maclaurin expansions. Derive the hyperbolic cosine of x, e^x, and tan x expansions, keeping the first three nonzero terms.

Apply Taylor and Maclaurin polynomials to express functions as series, use alternating series estimation theorem and Taylor inequality for error bounds, and approximate cube roots with a degree-two Taylor polynomial.