W11 Stepwise

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

01

01

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

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

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02

03

Finding a power series

Find a power series representation for these functions:

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

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03

01

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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04

04

Discovering the function from its Maclaurin series

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

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

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05

01

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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