I am stucked on this question and have no idea. Can somebody tell me how to solve this summation?
$\Sigma ^n_{k=1} 3^kk^2$
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2We need to know what you already know. For starters, do you know how to do $\sum^n_{k=1} 3^k$? – Ivan Neretin Mar 23 '24 at 08:40
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Do you know any basic differentiation? – Mark Bennet Mar 23 '24 at 08:41
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Yes, I know basics and summation of $3^k$. – Aospiyxmoq Mar 23 '24 at 08:48
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You can lots of ideas for this sort of summation from the answers to this very similar question: https://math.stackexchange.com/questions/30732 – joriki Mar 23 '24 at 09:03
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By the way, you can get displayed equations by enclosing them in double instead of single dollar signs. – joriki Mar 23 '24 at 09:03
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Hint. $$ \sum_{k=1}^nk^2x^k=x\frac{d}{dx}\left(x\frac{d}{dx}\sum_{k=1}^nx^k\right) $$ – Gonçalo Mar 23 '24 at 09:13
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Hi, welcome to Math SE. Alternative to @Gonçalo's hint: try the Ansatz $(An^2+Bn+C)3^n+Dn^2+En+F$. You can get all but one of the coefficients from the change under $n\mapsto n-1$. – J.G. Mar 23 '24 at 09:14