# Math Help - Sum of mixed series

1. ## Sum of mixed series

I have the series 1r, 2r^2,3r^3...(or in general nr^n)
what is the sum of this series? Im a bit confused as it is a mixture of an arithmetic series (1,2,3...) and a geometric (r^n)

2. ## Re: Sum of mixed series

Let $f(r) :=\sum_{n=1}^{+\infty}nr^n$. You can integrate it term-by-term, for example from $0$ to $x$, with $x<1$. Why?

3. ## Re: Sum of mixed series

Originally Posted by Schdero
I have the series 1r, 2r^2,3r^3...(or in general nr^n)
what is the sum of this series? Im a bit confused as it is a mixture of an arithmetic series (1,2,3...) and a geometric (r^n)

Is...

$\sum_{n=1}^{\infty} n\ r^{n} = r\ \sum_{n=1}^{\infty} n\ r^{n-1} = r\ \frac{d}{d r}\ \sum_{n=1}^{\infty} r^{n}$ (1)

... and, because for $|r|<1$...

$\sum_{n=1}^{\infty} r^{n} = \frac{1}{1-r} - 1$ (2)

... we have for $|r|<1$ ...

$\sum_{n=1}^{\infty} n\ r^{n} = \frac{r}{(1-r)^{2}}$ (3)

Kind regards

$\chi$ $\sigma$

4. ## Re: Sum of mixed series

Thank you for the quick reply! What if, e.g., im looking for the sum up to only n=8 though?

5. ## Re: Sum of mixed series

Originally Posted by Schdero
I have the series 1r, 2r^2,3r^3...(or in general nr^n)
what is the sum of this series? Im a bit confused as it is a mixture of an arithmetic series (1,2,3...) and a geometric (r^n)

You can get to a standard geometric series as follows....

$S_n=r+2r^2+3r^3+.....+nr^n$

$rS_n=r^2+2r^3+3r^4+....+nr^{n+1}$

$S_n-rS_n=r+r^2+r^3+....r^n-nr^{n+1}$

$S_n(1-r)=\left(r+r^2+r^3+...+r^n\right)-nr^{n+1}$

The geometric series in brackets can now be evaluated
and hence a closed form for the sum can be found.

6. ## Re: Sum of mixed series

Another way: denoting $S=r+2r^2+3r^3+\ldots\quad$ , we have $rS=r^2+2r^3+3r^3+\ldots$ . This implies

$(1-r)S=r+r^2+r^3+\ldots=\dfrac{1}{1-r}-1=\dfrac{r}{1-r}\quad (|r|<1)$

So, $S=\dfrac{r}{(1-r)^2}$

Edited: Sorry, I didn't see Archie Meade's post

7. ## Re: Sum of mixed series

Thank you very much to all of you!!