Evaluate
∞
∑ 1 / [n(n+1) ]
n=1
(hint: start with a geometric series, then use term-by-term integration)
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Attempt:
∞
∑ = 1/(1-x) for |x|<1
n=0
Take the indefinite integral of both sides...
∞
∑ /(n+1) + C= -ln(1-x)
n=0
Put x=0 => C=0
∞
∑ /(n+1) = -ln(1-x)
n=0
Divide both sides by x^2
∞
∑ /(n+1) = - ln(1-x) /
n=0
∞
∑ /(n+1) = - [ln(1-x) / ] - 1/x
n=1
How to continue?
Thanks for helping!