Problems

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 ${\displaystyle \frac{1}{5-x}}$ 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.

02

Power series of a derivative

Suppose that a function $f(x)$ has power series given by:

$$f(x)=x^2-\frac{x^4}{2}+\frac{x^6}{3}-\frac{x^8}{4}+\cdots =\sum _{n=0}^{\infty }{(-1)}^n\frac{x^{2n+2}}{n+1}$$

The radius of convergence of this series is $R=1$.

What is the power series of $f^{\prime }(x)$ and what is its interval of convergence?

03

Finding a power series

Find a power series representation for these functions:

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

04

Modifying and integrating a power series

(a) Modify the power series $\frac{1}{1-x}=1+x+x^2+x^3+\cdots ={\sum }_{n=0}^{\infty }{x}^n$ for $|x|<1$ to obtain the power series for $f(x)=\frac{1}{1+x^4}$.

(b) Now integrate this series to find the power series for $\int f(x)dx$.