Power series sum

Question

Find the radius of convergence and the sum of the series: $\sum \limits_{k=0}^\infty (2k^2+8k+5)x^k$.
I know how to find the radius of convergence but I have no idea how to calculate the sum of the series. Where do I begin? Thanks.

If you have found the radius of convergence, why don't you add that to the question? — Rushabh Mehta, Dec 15 '20 at 17:09
Try differentiating the power series $\sum_{k=0}^\infty x^k = \frac1{1-x}$. — player3236, Dec 15 '20 at 17:11
See https://math.stackexchange.com/questions/593996/how-to-prove-sum-n-0-infty-fracn22n-6/594019#594019 — lab bhattacharjee, Dec 15 '20 at 17:24

score 4 · Accepted Answer · answered Dec 15 '20 at 17:12

Hints:

For example,

$$f(x)=\sum_{k=0}^\infty x^k\,,\,\,|x|<1\implies f'(x)=\sum_{k=1}^\infty kx^{k-1}\implies f''(x)=\sum_{k=2}^\infty k(k-1)x^{k-2}\implies$$

$$f''(x)=\sum_{k=2}^\infty k^2x^{k-2}-\sum_{k=1}^\infty kx^{k-1}$$

but we know

$$f(x)=\frac1{1-x}\implies f'(x)=\frac1{(1-x)^2}\;\;\ldots\text{ and etc.}$$

score 2 · Answer 2 · answered Dec 15 '20 at 18:04

Hint:

You can rewrite the sums as \begin{align} \sum_{k=0}^\infty 8k x^{k}&=8x\sum_{k=0}^\infty k x^{k-1},\\[1ex] \sum_{k=0}^\infty 2k^2x^k&=2x^2\sum_{k=0}^\infty k(k-1)x^{k-2}+2x\sum_{k=0}^\infty kx^{k-1}, \end{align} whence $$\sum \limits_{k=0}^\infty (2k^2+8k+5)x^k=2x^2\sum_{k=0}^\infty k(k-1)x^{k-2}+10x\sum_{k=0}^\infty kx^{k-1}+5\sum_{k=0}^\infty x^{k}.$$

Power series sum

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