What is the sum of the series 1/3 + 2/9 + 3/27 + 4/81 + ........

I remember solving this in highschool , but now I don't remember how to find sum of these kind of series .

I want to find the sum of the general series

Sum $\sum_{n=1}^{\infty} n .a^{-n} = ?$

and $\sum_{n=1}^{N}n. a^{-n} = ?$

Answer

Here is an approach that relies on the relationships (i) $k=\sum_{\ell=1}^k(1)$ and (ii) $\sum_{k=\ell}^N r^k=\frac{r^{\ell}-r^{N+1}}{1-r}$ for $|r|<1$. Then, with $r=3^{-1}$ we have

$$\begin{align} \sum_{k=1}^N \frac{k}{3^k}&=\sum_{k=1}^N 3^{-k}\sum_{\ell =1}^k(1)\\\\ &=\sum_{\ell =1}^N \sum_{k=\ell}^N 3^{-k}\\\\ &=\sum_{\ell =1}^N \frac{3^{-\ell}-3^{-(N+1)}}{1-1/3}\\\\ &=\frac32 \sum_{\ell =1}^N \left(3^{-\ell}-3^{-(N+1)}\right)\\\\ &=\frac 32\left(\frac{3^{-1}-3^{-(N+1)}}{1-1/3}\right)-\frac N2 3^{-N}\\\\ &=\frac34 -\frac34 3^{-N}-\frac12 N3^{-N} \end{align}$$

Note as $N\to \infty$ the sum of interest approaches $3/4$.