Question

Find the values of x for which the series converges, and for these values find its sum.

x - x^3 + x^5 - x^7 + x^9 - ...

Answer 1

\[\sum_{n=0}^\infty (-1)^n x^{2n+1}=x-x^3+x^5-x^7+\cdots\]

use ratio test on the term of the series

Answer 2

ok thanks ya :D

Answer 3

This serie always converge, right?

Answer 4

don't know..

Answer 5

Ratio Test for \[\sum u_n\]
\[\lim_{n\rightarrow \infty}\left|\frac{u_{n+1}}{u_n}\right|<1\]
here, \[u_n=(-1)^nx^{2n+1}\] and \[u_{n+1}=(-1)^{n+1}x^{2(n+1)+1}\]

Answer 6

Ok! Not always converge, x must be in some range of values, thanks.

Answer 7

\[\left|\frac{u_{n+1}}{u_n}\right|=\left|x^2\right|\]
ratio test: \[\lim_{n\rightarrow \infty}|x^2|<1\Rightarrow -1<x<1\]

Answer 8

For the sum, you can fin this usefull,
\[S=x−x^3+x^5±…\]
\[x^2S=x^3−x^5+x^7±…\]
\[S+x^2S=x⇒S=\frac{x}{1+x^2},\ \ \ \ ∀x∈(−1,1)\]