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**:

**Section 8.1**– Introduction – The Taylor Approximation**Section 8.2**– Power Series and Analytic Functions**Section 8.3**– Power Series Solutions to Linear Differential Equations**Section 8.4**– Equations with Analytic Coefficients

A **power function** is a function in the form .

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

where are the coefficients of the power series.

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

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

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

The series converges only when . [].

The series converges for all . [].

There is a positive number such that the series converges if and diverges if .

**NOTE**: If , then you need to verify the convergence at each endpoint, and , leading to the following possible intervals of convergence:-
where is the
**Radius of Convergence**of the power series.

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

- Rewrite the function as a power series. Explain.

Rewrite the following functions as power series:

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

Rewrite the following functions as power series:

- A series
**converges**if - A series
**diverges**if - The ratio test is
**inconclusive**if

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

- Evaluate
- Evaluate
- Evaluate
- Evaluate

Suppose is differentiable on its interval of convergence, then

Suppose is integrable on its interval of convergence, then

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

Suppose , then

Shift the following summation indices so that the series starts at

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

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

.

Definition: Maclaurin Series

A **Maclaurin series** is a Taylor series when .

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

An **odd** number has the form for any .

Here are some examples:

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