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Ac3
 one year ago
use equation 1 to find a power series representation for f(x)=ln(1x). What is the radius of convergence?
Ac3
 one year ago
use equation 1 to find a power series representation for f(x)=ln(1x). What is the radius of convergence?

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anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Not sure what you mean by equation (1), but I'll take a wild guess and suppose it's \[\frac{1}{1x}=\sum_{n=0}^\infty x^n\quad\text{for }x<1\] Notice that \[\frac{d}{dx}\ln(1x)=\frac{1}{1x}=\sum_{n=0}^\infty x^n\] Integrating, you have \[\ln(1x)=\sum_{n=0}^\infty \frac{x^{n+1}}{n+1}+C\] If we consider \(x=0\), you would find that \[\ln(10)=\sum_{n=0}^\infty \frac{0^n}{n+1}+C~\implies~C=0\]

Ac3
 one year ago
Best ResponseYou've already chosen the best response.0Alright so we have \[\sum_{0}^{\infty} \frac{ x ^{n+1} }{ n+1 }\]

Ac3
 one year ago
Best ResponseYou've already chosen the best response.0using that do we now we use that to find the power series of xln(1x)

Ac3
 one year ago
Best ResponseYou've already chosen the best response.0@SithsAndGiggles Would I just multiply the whole thing by x getting. \[\sum_{n=0}^{\infty} \frac{ x ^{n+2} }{ n+1 }\]

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Right, if the power series for \(f(x)\) is given by \(S\), then the power series for \(x f(x)\) is \(xS\). You're missing the minus sign, btw.

Ac3
 one year ago
Best ResponseYou've already chosen the best response.0we're on the last part of the entire question so now by putting x=1/2 in your result from part a (that's our first one), express ln2 as te sum of an infintie series.

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Okay, so if \(x=\dfrac{1}{2}\), then \[\ln\left(1\frac{1}{2}\right)=\ln\frac{1}{2}=\ln2^{1}=\ln2\]

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Yep! The series representation is simple enough, you're just replacing \(x\) with \(\dfrac{1}{2}\). \[\ln2=\sum_{n=0}^\infty \frac{\left(\frac{1}{2}\right)^{n+1}}{n+1}\]which you can rewrite in several ways.

Ac3
 one year ago
Best ResponseYou've already chosen the best response.0Thank you dude your freaking awesome!!!
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