How can you rewrite the function as a power series?
f(x) = x/ (1 + 2x^2)
I know I have to take out the x so x ( 1 / (1 + 2x^2)
however I'm stuck afterwards

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- anonymous

- schrodinger

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- Jhannybean

- anonymous

\[\frac{1}{1-x}=1+x+x^2+x^3+...\]
\[\frac{1}{1+x}=1-x+x^2-x^3+...\]
\[\frac{1}{1+2x^2}=1-2x^2+(2x^2)^2-(2x^2)^3+...\]

- anonymous

So would I write the sum as \[\sum_{n=1}^{\infty} (-1)^{n} (x)^{2n+1}\]

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## More answers

- anonymous

you need a 2 inside the parentheses

- anonymous

Oh yeah my bad. But how can I find the radius of convergence if I can't use the ratio test afterwards? I mean there's no division going on. Only thing I can think of is the root test but its negative, so the root test won't work perfectly.

- anonymous

I thought power series was just sum x^n

- anonymous

well the representation

- anonymous

Well I took the x and added it to the 2x^(2n+1) because the x was "outside" of the sum. So i just added it and thats what I did. I don't feel like writting the equation. I don't know if you follow what I'm saying.It was my guess really.

- anonymous

converges if \(|2x|<1\) i.e if \(|x|<\frac{1}{2}\)

- anonymous

Ah So i just take it right away. Makes sense. Thanks a lot^^

- anonymous

also you should start your summation at \(n=0\) i think

- anonymous

to be more precise i guess if converges if \(|2x^2|<1\) but same answer

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