MATH 105 101
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Week 10 Learning Goals
Overview
Last week, we started on infinite series. In particular, we learnt about two tests of convergence and/or divergence: namely, the divergence test and the integral test. This week, we
will learn a few more such tests designed to verify convergence or divergence of an infinite
series, even when we cannot evaluate the series exactly. We will then begin the study of a
very special class of functions whose functional forms are given by infinite series.
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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. State the Ratio Test (Theorem 8.14 p.641), Root Test (Theorem 8.15 p.642), Comparison Test (Theorem 8.16 p.643) and Limit Comparison Test (Theorem 8.17 p.645) for
convergence or divergence of an infinite series with positive terms. [Conceptual]
2. Use the tests learnt so far to check if an infinite series converges or diverges. Be
able to recognize forms that indicate the applicability or otherwise of certain tests.
[Procedural]
3. Given the definition of absolute and conditional convergence (Definition p.654). Give
examples of some absolutely convergent and some conditionally convergent series. We
do NOT cover Remainders in Alternating series.
4. State the Alternating Series Tests (Theorem 8.18 p.650), and apply it to determine
convergence of alternating series.
5. Given a function f that can be differentiated up to any order, describe the n-th order
Taylor polynomial for f with center a.[Procedural]
Example problem: Find the 6-th order Taylor polynomial for f (x) = sin(x) centered at
x = π.
6. Given a function f that can be differentiated up to any order, explain why the n-th
order Taylor polynomial may be considered a good approximation to f . [Conceptual]
7. Define a power series, and the radius of convergence of a power series (Definition p.676).
Use the radius of convergence to determine an interval where the series converges.
8. Given a power series, be able to find the radius of convergence of a power series.
[Procedural]
∞
X
(2k)!
Example problem: Find the radius of convergence of the power series
(x − 4)k .
2
k
k=1
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MATH 105 101
Week 10 Learning Goals
9. Be able to manipulate power series algebraically, such as sums, differences, multiplication by a power, composition, integration, and differentiation. [Procedural]
Example problem: Find the power series representation of arctan(x2 ) centered at 0.
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