I am given three summations as follows:

$S_0 = \sum_{k=0}^n x^k$

$S_1 = \sum_{k=1}^n kx^k$

$S_2 = \sum_{k=1}^n k^2x^k$

The question is how do I express $S_2$ in terms of $S_1$ and $S_0$?

it is given that n is in natural numbers, x is real, and x is not equal to 1.

I am struggling to understand how to approach this problem. The first step that I took is reindexing so that they all begin with k=0, since that term would be 0 for $S_1$ and $S_2$. Also I realize that these are all related by increasing the exponent of k. Unfortunately, I am unsure what to do with this information. Any hints on where to start would be appreciated!

  • $\begingroup$ I think the problem is about how to write $S_1$ and $S_2$ in terms of $S_0$. If this is the case try manipulations with derivatives. The derivative of a summation increase in 1 the starting position in the index. $\endgroup$ – Masacroso Nov 30 '15 at 0:58
  • $\begingroup$ My mistake. The question is how to write $S_2$ in terms of $S_1$ and $S_0$ $\endgroup$ – Taylor Nov 30 '15 at 1:47
  • $\begingroup$ It is the same, use derivatives. Write $S_0$ in terms without the summation symbol and derive it ($\frac{\rm d}{\rm dx}S_0$) and see what happen. $\endgroup$ – Masacroso Nov 30 '15 at 3:03

You can write $S_0$ as $\frac{1-x^{n+1}}{1-x}$ (are you OK with that?) and then rewrite $S_1$ as

$S_1 = \sum_{k=1}^{n} k x^k = \frac{1}{x} \sum_{k=1}^{n} k x^{k-1}$. You can then use $\sum_{k=1}^{n} k x^{k-1} = \frac{d S_0}{d x}$.

For $S_2$ use $\frac{d S_1}{d x} = \sum_{k=1}^{n} k^2 x^{k-1} = \frac{1}{x} S_2$.


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