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How to find an explicit formula for the term $$\sum^n_{k=1}\frac{k^2}{2^k}$$ Then I discovered that
$$\sum^n_{k=1}\frac{k^2}{2^k}=1^22^n+2^22^{n-1}+3^22^{n-2}+\cdots+n^22^1-\frac{n(n+1)(2n+1)}{6}$$ But how to write $1^22^n+2^22^{n-1}+3^22^{n-2}+\cdots+n^22^1$ into explicit formula?? $$\\$$ A hint would be grateful.


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marked as duplicate by Antonio Vargas, Did sequences-and-series Oct 27 '15 at 16:39

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  • $\begingroup$ Hint: Differentiate once and twice the series $$\sum_{n=0}^\infty x^n=\frac1{1-x}$$ $\endgroup$ – Did Oct 27 '15 at 16:34
  • $\begingroup$ See also this question and this question. $\endgroup$ – Antonio Vargas Oct 27 '15 at 16:39