Question

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

Answer 1

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

Answer 2

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

Answer 3

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

Answer 4

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

Answer 5

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