APEX Calculus - Sequences and Series

APEX Calculus - Sequences and Series

Explore an in-depth 8-hour course on sequences and series in calculus. Begin with an introduction to sequences, learning how to list terms, find formulas, and understand limits. Dive into limit properties, bounded sequences, and monotone sequences with practical examples. Progress to series, covering divergent series, geometric series, p-series, and telescoping series. Master integral and comparison tests, including limit comparison. Study the ratio and root tests for series convergence. Examine alternating series, absolute convergence, and approximation theorems. Delve into Taylor polynomials, understanding their coefficients, definitions, and error estimates. Investigate power series, exploring their radius of convergence, derivatives, and integrals. Conclude with an extensive look at Taylor series, including Maclaurin series, natural logarithms, binomial series, and techniques for manipulating known series. Apply your knowledge to solve initial value problems and integrate series.