Evaluate

∞

∑ 1 / [n(n+1) $\displaystyle 5^n$]

n=1

(hint: start with a geometric series, then use term-by-term integration)

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Attempt:

∞

∑ $\displaystyle x^n$ = 1/(1-x) for |x|<1

n=0

Take the indefinite integral of both sides...

∞

∑ $\displaystyle x^{n+1}$ /(n+1) + C= -ln(1-x)

n=0

Put x=0 => C=0

∞

∑ $\displaystyle x^{n+1}$ /(n+1) = -ln(1-x)

n=0

Divide both sides by x^2

∞

∑ $\displaystyle x^{n-1}$ /(n+1) = - ln(1-x) / $\displaystyle x^2$

n=0

∞

∑ $\displaystyle x^{n-1}$ /(n+1) = - [ln(1-x) / $\displaystyle x^2$] - 1/x

n=1

How to continue?

Thanks for helping!