Power Series Solutions

Power Series SolutionsPrerequisitesDefinition: Power FunctionDefinition: Power SeriesDefinition: Power Series Centered at Definition: Radius of Convergence of Power Series:Definition: Interval of Convergence of a Power SeriesTheoremInvestigation 01Investigation 02Investigation 03Investigation 04Investigation 05Series ConvergenceRatio TestInvestigation 06Differentiation of Power SeriesDefinition: Derivative of a Power SeriesIntegration of Power SeriesDefinition: Integration of a Power SeriesInvestigation 06Shifting the Summation Index of a Power SeriesDefinition: Index Shift of a Power SeriesInvestigation 07Investigation 08Taylor SeriesDefinition: Taylor SeriesMaclaurin SeriesCommon Maclaurin SeriesOdd and Even NumbersProduct NotationCC BY-NC-SA 4.0

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

Prerequisites

Definition: Power Function

A power function is a function in the form .

Definition: Power Series

A power series is a sum of power functions: for some in some interval.

Definition: Power Series Centered at

where are the coefficients of the power series.

Definition: Radius of Convergence of Power Series:

If where is a positive real number, then the radius of convergence of the power series is .

Definition: Interval of Convergence of a Power Series

The Interval of Convergence of a Power Series is the interval of all -values for which the series converges.

Theorem

For a given power series , the possible Interval of Convergence of a Power Series are

Investigation 01

  1. Identify the type of series
  2. Determine for what values of , the series converges. Explain.
  3. Find the total sum of the convergent series. Explain.

Investigation 02

  1. Rewrite the function as a power series. Explain.

Investigation 03

Rewrite the following functions as power series:

Investigation 04

  1. Write out the first five (5) terms of
  2. Differentiate the first five (5) terms of
  3. Use the pattern to write the derivative from Step 2 as a series
  4. Integrate the first five (5) terms of
  5. Use the pattern to write the derivative from Step 4 as a series
  6. Make a general conclusion about the differentiation and integration of power series.

Investigation 05

Rewrite the following functions as power series:

Series Convergence

Ratio Test

Investigation 06

Determine the interval and radius of convergence for each of the following power series:

  1. Evaluate
  2. Evaluate
  3. Evaluate
  4. Evaluate

Differentiation of Power Series

Definition: Derivative of a Power Series

Suppose is differentiable on its interval of convergence, then

Integration of Power Series

Definition: Integration of a Power Series

Suppose is integrable on its interval of convergence, then

Investigation 06

Find the derivative and antiderivative for each of the following power series:

Shifting the Summation Index of a Power Series

Definition: Index Shift of a Power Series

Suppose , then

Investigation 07

Shift the following summation indices so that the series starts at

Investigation 08

Shift the following summation indices to combine the following series [i.e., must have same exponent]:

Taylor Series

Definition: Taylor Series

Suppose is differentiable times at , then the Taylor series, a.k.a., Taylor approximation, a.k.a., Taylor polynomial is

.

Maclaurin Series

Definition: Maclaurin Series

A Maclaurin series is a Taylor series when .

Common Maclaurin Series

Odd and Even Numbers

An even number has the form for any integer , i.e., .

An odd number has the form for any .

Product Notation

Here are some examples:

CC BY-NC-SA 4.0

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License [http://creativecommons.org/licenses/by-nc-sa/4.0/].

Last Modified: Wednesday, 10 November 2020 18:26 EDT