## anonymous 5 years ago Show the Series n=1 to infinity nX^n converges to f(x)=x/(1-x)^2... without the use of taylors theorem

1. anonymous

take the derivative of $\frac{1}{1-x}=\Sigma x^n$ to get $\frac{1}{(1-x)^2}=\Sigma n x^{n-1}$

2. anonymous

then multiply by x to get $\frac{x}{(1-x)^2}=\Sigma nx^n$

3. anonymous

of course you have to be able to justify taking the derivative term by term, and also note that the radius of convergence is -1 < x < 1

4. anonymous

Thanks Yeah I did the term by term derivative, so is the initial geo series starting at n=1?

5. anonymous

the initial series starts at n = 0 not n = 1, but this is not a problem because the final one gives 0 for n = 0 so you might as well start at n = 1