Thread: Problem 30
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Old July 9th, 2007, 02:55 PM
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Let S_n=\sum_{k=1}^nkx^{k-1}
We have S_n=1+2x+3x^2+\ldots +nx^{n-1} (1).
Multiplying (1) with x we have
xS_n=x+2x^2+3x^3+\ldots +(n-1)x^{n-1}+nx^n (2)
Substracting (2) from (1) yields
S_n(1-x)=1+x+x^2+\ldots +x^{n-1}-nx^n=\frac{1-x^n}{1-x}-nx^n\Rightarrow S_n=\frac{1-x^n}{(1-x)^2}-\frac{nx^n}{1-x}
Then \sum_{n=1}^{\infty}nx^{n-1}=\lim_{n\to\infty}S_n=\frac{1}{(1-x)^2}
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