# 1 Overview ```MATH 105 101
1
Week 11 Learning Goals
Overview
We continue our study of power series, focusing on the relation between known functions
and their power series representations. We also look at some applications of power series.
2
Learning objectives
By the end of the week, having participated in lectures, worked through the indicated sections
of the textbook and other resources, and done the suggested problems, you should be able
to:
1. Given a function f that can be differentiated up to any order, write down the Taylor
series for f centered at x = a. Define the Maclaurin series of f (Definition p.685).
[Conceptual/Procedural]
Example problem: Find the Maclaurin series for the function f (x) = ex . Determine
its radius of convergence.
2. Use Taylor series to evaluate limits. [Procedural]
ln(1 + x) − x − x2 /2
.
x→0
x3
Example problem: Evaluate lim
3. Manipulate Taylor series of known functions to obtain Taylor series for other functions.
[Procedural]
2
Example problem: Find the Maclaurin series for the function f (x) = e(x ) . Use this to
determine the value of f (100) (0).
4. Evaluate a power series in terms of a known function. Use this to evaluate certain
numerical series. [Procedural]
∞
X
1
Example problem: Evaluate
.
k
π k!
k=1
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