How do I calculate this sum, using power series? $$\sum_{n\ge0}{{\frac{(n+1)(n+2)}{3^n}}}$$ I want just some hints to get me on the right track
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$\begingroup$ Looks the the second derivative of a power series in x evaluated at 1. $\endgroup$– Charlie FrohmanNov 24, 2016 at 18:31
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$\begingroup$ math.stackexchange.com/questions/593996/… $\endgroup$– lab bhattacharjeeNov 24, 2016 at 18:31
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1 Answer
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Hint:
$$(n+2)(n+1)x^n$$
is the second derivative of $x^{n+2}$. Therefore,
$$\sum_{n \ge 0} (n+2)(n+1)x^n \tag{1}$$
is the second derivative of $$ \sum_{n \ge 0} x^{n+2} = \frac{x^2}{1-x}. \tag{2} $$
See if you can use (2) to get a closed form for (1). Then plug in a specific value of $x$.
answered Nov 24, 2016 at 18:31