Remainder Theorem

The n-th Talor polynomial The polynomial is called the n-th Taylor polynomial for f about c

The n-th Maclaurin polynomial The polynomial is called the n-th Maclaurin polynomial for f

Taylor formula for f with the Remainder The difference is called the n-th remainder for the Taylor series of f about c.

Thr Remainder Theorem Assume that all the first n+1 derivatives exist on an interval l containing c and

Example (1) Approximate e to five decimal places

Why we chose n=9 ?

Example (2) Approximate sin3 ○ to five decimal places

Approximating the sum of a convergent alternating series

Theorem Let T be the sum of an alternating series satisfying the main convergence test for an alternating series. Then: 1. T lies between any two successive partial sums of the series 2. If T is approximated by a partial sum T n, then: a. |T - T n | ≤ a n+1 b. The sign of the error is the same as that of the coefficient of a n+1

Using power series to approximate definite integrals

Example Give an estimation of the integral on the right accurate to 3 decimal places

A power series for the given integral

Approximating the integral

Homework