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Proving

jazmin_afc

Does anyone know how to prove the following:

sum n=1 to infinity from nx^n = x/(1-x)^2

Reply 1

an_atheist

What is the general formula for the summation of a geometric series?

Reply 2

joostan

Original post by jazmin_afc

Does anyone know how to prove the following:

sum n=1 to infinity from nx^n = x/(1-x)^2

Depending on what you're allowed to assume, you could observe that:

\displaystyle\sum_{n=1}^{ \infty} nx^n=x\displaystyle\sum_{n=1}^{ \infty} nx^{n-1}

.
Clearly the sum on the right hand side looks like a derivative, of what is in fact a familiar series too, though issues of convergence might be important depending on what kind of proof this is.

Alternatively you could perform partial fractions on the right hand side and expand the series there.