Two Quick Questions - Power Series

Sprintz · #1 $Old$ November 3rd, 2009, 08:19 AM

Hello all; having some trouble with these last homework questions!

Instructions: Find a power series representation for the function and determine the radius of convergence.

#1) f[x] = x^2/(1-2x)^2

Should I use partial fractions to split this up?

#2) Evaluate the indefinite integral as a power series. What is the radius of convergence. Integral of ArcTan(x^2)dx

Thanks so much for your help in the past and future all!

chisigma · #2 $Old$ November 3rd, 2009, 09:32 AM

Because ...

$\frac{1}{1-2x}= \sum_{n=0}^{\infty} (2x)^{n}$ , $|x|<\frac{1}{2}$ (1)

... and...

$\frac{d}{dx} \frac{1}{1-2x} = \frac{2}{(1-2x)^{2}}$ (2)

... is...

$\frac{x^{2}}{(1-2x)^{2}}= \sum_{n=0}^{\infty} \frac{2^{n-1}}{n+1}\cdot x^{n+3}$ , $|x|<\frac{1}{2}$ (3)

Kind regards

$\chi$ $\sigma$

tonio · #3 $Old$ November 3rd, 2009, 09:34 AM

Quote:

Originally Posted by Sprintz $View Post$

Hello all; having some trouble with these last homework questions!

Instructions: Find a power series representation for the function and determine the radius of convergence.

#1) f[x] = x^2/(1-2x)^2

Should I use partial fractions to split this up?

As $\frac{1}{1-2x}=1+2x+4x^2+...+(2x)^n+...\,\,whenever\,\,|x|<\frac{1}{2}\,,\,\,and\,\,since\,\,$ $\frac{2}{(1-2x)^2}=\left(\frac{1}{1-2x}\right)'\,,\,\,then...$

#2) Evaluate the indefinite integral as a power series. What is the radius of convergence. Integral of ArcTan(x^2)dx

$\arctan x=x-\frac{x^3}{3}+\frac{x^5}{5}-...\,,\,\,so \,\,\arctan x^2=x^2-\frac{x^6}{3}+...$

Tonio

Thanks so much for your help in the past and future all!

.