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gokugreene

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**Infinite Series Calculus 2 homework problem**

Hello. I need help finding the answer on two of these.

#1. Find a power series representation for [tex]f(x)=ln(1+x^2)[/tex]

I have

[tex]f'(x)=\frac{(2x)}{(1+x^2)}[/tex]

So I factor out 2x and have [tex]\frac{1}{(1+x^2)}[/tex] which is a geometric series =

[tex]\sum(-1)^{n}x^{2n}[/tex] then multiply 2x times the series

2x * [tex]\sum(-1)^{n}x^{2n}[/tex] = [tex]\sum(-1)^{n}2x^{2n+1}[/tex]

Then, [tex]\int\sum(-1)^{n}2x^{2n+1}[/tex] = [tex] \sum\frac{(-1)^{n+1}2x^{2n+2}}{2n+2}[/tex]

Did I do that right??

Ok Problem #2 Find the radius of convergence and interval of convergence of the series.

[tex]\sum\frac{(-2)^{n}(x+3)^{n}}{\sqrt{n}}[/tex]

for R I get 1/2 and I am having trouble figuring out whether the endpoints (5/2)<x<(7/2), are convergent.

Thanks!

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