## anonymous 3 years ago 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

1. anonymous

@terenzreignz

2. anonymous

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

3. anonymous

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

4. anonymous

you need a 2 inside the parentheses

5. 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.

6. anonymous

I thought power series was just sum x^n

7. anonymous

well the representation

8. 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.

9. anonymous

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

10. anonymous

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

11. anonymous

also you should start your summation at $$n=0$$ i think

12. anonymous

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