- #1
azatkgz
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Homework Statement
Find the sum of the following series
[tex]\sum_{n=1}^{\infty}n(n+1)x^n[/tex]
The Attempt at a Solution
[tex]x\sum_{n=1}^{\infty}n(n+1)x^{n-1}[/tex]
[tex]x\int_{0}^{x}(\int_{0}^{x}f(t)dt)dt=x(x^2+x^3+x^4+x^5+\cdots)=x\frac{x^2}{1-x}[/tex]
[tex]x\frac{d^2}{dx^2}\frac{x^2}{1-x}=\frac{2x}{1-x}[/tex]