W11 Stepwise

Due date: Thursday 3/26, 11:59pm

Power series as functions

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Modifying geometric power series

Consider the geometric power series $\frac{1}{1-x}=1+x+x^2+x^3+\cdots ={\sum }_{n=0}^{\infty }{x}^n$ for $|x|<1$.

For this problem, you should modify the series for $\frac{1}{1-x}$.

(a) Write ${\displaystyle \frac{1}{5-x}}$ as a power series and determine its interval of convergence.

(b) Write ${\displaystyle \frac{1}{16+2x^3}}$ as a power series and determine its interval of convergence.

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Solution

Solutions---0230-01

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Finding a power series

Find a power series representation for these functions:

(a) $f(x)={\displaystyle \frac{x^2}{x^4+81}}$ $$ (b) $g(x)=x^2\ln (1+x)$

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Solution

Solutions---0230-03

Taylor and Maclaurin series

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Maclaurin series

For each of these functions, find the Maclaurin series, and the interval on which the expansion is valid.

(a) $x\ln (1-5x)$ $$ (b) $x^2\cos (x^3)$

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Solution

Solutions---0240-01

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Discovering the function from its Maclaurin series

For each of these series, identify the function of which it is the Maclaurin series.

(a) ${\displaystyle \sum _{n=0}^{\infty }{(-1)}^n{2}^n{x}^n}$ $$ (b) ${\displaystyle \sum _{n=0}^{\infty }{(-1)}^n\frac{x^{3n}}{n!}}$

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Solution

Solutions---0240-04

Applications of Taylor series

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Approximating $1/e$

Using the series representation of $e^x$, show that:

$$\frac{1}{e}=\frac{1}{2!}-\frac{1}{3!}+\frac{1}{4!}-\cdots$$

Now use the alternating series error bound to approximate $\frac{1}{e}$ to an error within ${10}^{-3}$.

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Solution

Solutions---0250-01