## Series and Approximations

**Excel Exercise**

**Series and Convergence**

**Series**

**Convergence Tests**

**Visual Calculus Sequences & Series**

**Visual Calculus Power Series**

**Visual Calculus Taylor Polynomials**

A Gallery of Animations of Taylor Polynomials

Another Visual Approach to Taylor Polynomials

**AP Calculus (BC) Topics**

**IV. Polynomial Approximations and Series**

**Concept of Series**. Series are defined as a sequence of partial sums, and convergence is defined as the limit of the sequence of partial sums. Technology is used to explore convergence or divergence of various examples.

**Series of Constants**. Motivating examples including decimal expansion. Geometric series with applications. The harmonic series. Alternating series with error bound. Terms of series as areas of rectangles and their relationship to improper integrals, including the integral test and its use in testing convergence of p-series. Comparing series to test for convergence or divergence.

**Taylor Series**. Taylor polynomial approximation with graphical demonstration of convergence. (For example, viewing graphs of various Taylor polynomials of the sine function approximating the sine curve.) The general Taylor series centered at x = a. Formal manipulation of Taylor series and shortcuts to computing Taylor series, including differentiation, antidifferentiation, and the formation of new series from known series. Functions defined by power series and radius of convergence. Lagrange error bound for Taylor polynomials.