Let ( r=1+\dfrac{1}{n} ). Series to (n) terms: [ S=\sum_{k=1}^{n} k, r^{k-1} ]
Using (\sum_{k=1}^{n}kx^{k-1}=\dfrac{1-(n+1)x^n+nx^{n+1}}{(1-x)^2})
With (x=1+\frac1n), after simplification one obtains (S=n^2).
Proved.
11TH•Mathematics•Sequences And Series•Question 10
Overview
Prove that $1 + 2\left(1+\dfrac{1}{n}\right) + 3\left(1+\dfrac{1}{n}\right)^2 + \cdots$ to $n$ terms $= n^2$.
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