Power Series

Author: John J Weber III, PhD Corresponding Textbook Sections:

Expected Educational Results

Bloom’s Taxonomy

A modern version of Bloom’s Taxonomy is included here to recognize various different levels of understanding and to encourage you to work towards higher-order understanding (those at the top of the pyramid). All Objectives, Investigations, Activities, etc. are color-coded with the level of understanding.

Bloom’s Taxonomy for different levels of understanding
Figure 1.1: Bloom's Taxonomy

Power Series

Derivatives and Antiderivatives of Power Series

Definition: Derivative of a Power Series

ddx[n=0cn(xa)n]=n=1ncn(xa)n1

NOTE: Differentiating a power series shifts the index of the series.

Definition: Integral of a Power Series

[n=0cn(xa)n]dx=C+n=0cn(xa)n+1n+1

NOTE: Remember: Antiderivatives require the +C.

Investigation 07

Find the derivatives of the following power series:

  1. n=1xnn

  2. n=1xnn3

  3. n=0(1)n4nxn

  4. n=0(1)n(x3)n2n+1

  5. n=1(2x1)n5nn

Investigation 08

Find the antiderivatives of the following power series:

  1. n=1xnn

  2. n=1xnn3

  3. n=0(1)n4nxn

  4. n=0(1)n(x3)n2n+1

  5. n=1(2x1)n5nn

Investigation 09

Find the power series for f(x):

  1. f(x)=tan1(x)

  2. f(x)=ln(1+x)

Investigation 10

Find the power series for f(x):

  1. f(x)=tan1(x)

  2. f(x)=ln(1+x)

Investigation 11

Find the power series for f(x)dx:

  1. f(x)=1(1+x)2

Investigation 12

Find the power series for f(x):

  1. f(x)=1(1+x)2

Homework

At this time, you should be able to complete the following assignments:

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Created: Tuesday, 1 December 2020 7:38 EDT Last Modified: Tuesday, 30 May 2023 – 23:12 (EDT)