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We have the next sequence: $$\sum_{t=1}^{n}\frac{t^2}{(1+r)^t}$$ The problem is: find the partial sum of the sequence $S_n$. Should I consider above mentioned sequence as "Arithmetico-Geometric Sequence"?Could someone show me the right way? Thanks in Advance!

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This is $\sum_{t=1}^n t^2x^t$ where $x=1/(1+r)$. Now $$\sum_{t=1}^n t^2x^t=x\frac d{dx}\sum_{t=1}^n tx^t$$ and $$\sum_{t=1}^n tx^t=x\frac d{dx}\sum_{t=1}^nx^t$$ etc.

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  • $\begingroup$ Is $\frac{d}{dz}$ partial derivative and what you have denoted by $z$? $\endgroup$
    – David
    Jun 15 '17 at 6:17

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