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What functions are represented by the following power series? $$\sum\limits_{k=1}^{\infty}kz^k \quad \quad \quad \sum\limits_{k=1}^{\infty}k^2z^k$$

Would this involve using a Taylor expansion? I cannot think of any general functions that follow a similar representation (ie. $e^x$ involves factorials).

I wonder if I need to be considering a differentiation of a general $\sum\limits_{k=1}^{\infty}z^k$, since the two functions seem to follow that format? However the indices don't tally, as a differentiation of the general term above would be $\sum\limits_{k=1}^{\infty}kz^{k-1}$.

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Let $f(z)=\frac1{1-z}=\sum_{k=0}^\infty z^k$, then $$\sum_{n=1}^\infty kz^k=z\sum_{n=1}^\infty kz^{k-1}=zf'(z).$$

$$f''(z)=\sum_{k=2}^\infty k(k-1)z^{k-2}=\sum_{k=2}^\infty k^2z^{k-2}-\sum_{k=2}^\infty kz^{k-2}.$$

So, $$\sum_{k=2}^\infty k^2z^{k-2}=f''(z)+\frac{f'(z)}{z},$$ and $$\sum_{k=2}^\infty k^2z^k=z^2f''(z)+zf'(z).$$

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Ins the same spirit as Tim Raczkowski's answer, consider the case of $$S_p=\sum_{n=0}^\infty k^p z^k$$ and now write $$k^p=Ak(k-1)(k-2)\cdots (k-p)+Bk(k-1)(k-2)\cdots (k-p+1)+$$ $$Ck(k-1)(k-2)\cdots (k-p+2)+Dk(k-1)(k-2)\cdots (k-p+3)+\cdots$$ and identify the coefficients $A,B,C,D,\cdots$.

For example $p=5$ will give $A=1$, $B=10$, $C=25$, $D=15$, $E=1$ and so $$S_5=z^5\sum_{n=0}^\infty k(k-1)(k-2)(k-3)(k-4) z^{k-5}+$$ $$10 z^4\sum_{n=0}^\infty k(k-1)(k-2)(k-3) z^{k-4}+$$ $$25 z^3\sum_{n=0}^\infty k(k-1)(k-2) z^{k-3}+$$ $$15 z^2\sum_{n=0}^\infty k(k-1) z^{k-2}+$$ $$ z\sum_{n=0}^\infty k z^{k-1}$$ in which you easily recognize the successive derivatives of $\sum_{n=0}^\infty z^{k}=\frac{1}{1-z}$.

I am sure that you can take from here.

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