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 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.

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.

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.

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.

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.

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

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.

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.