<s>[INST] Finding the infinite series: $3 \cdot \frac{9}{11} + 4 \cdot \left(\frac{9}{11}\right)^2 + 5 \cdot \left(\frac{9}{11}\right)^3 + \cdots$? I'm trying to find the infinite sum that is defined by:
$$
3 \cdot \frac{9}{11} + 4 \cdot \left(\frac{9}{11}\right)^2 + 5 \cdot \left(\frac{9}{11}\right)^3 + \cdots
$$
However, I do not know of any known formula to do this. Am I missing something really simple? Thanks!
 [/INST] For $|r|<1$ we have $$\sum_{k=0}^{\infty}r^k=\frac{1}{1-r}$$
Then, by taking derivatives respect to $r$
$$\sum_{k=0}^{\infty}kr^{k-1}=\frac{1}{(1-r)^2}\quad\implies\quad \frac{1}{r}+2+\sum_{k=3}^{\infty}kr^{k-2}=\frac{1}{r(1-r)^2}$$
Then
$$\sum_{k=3}^{\infty}kr^{k-2}=\frac{1}{r(1-r)^2}-\frac{1}{r}-2$$
Thus
$$\sum_{k=3}^{\infty}k\left(\frac{9}{11}\right)^{k-2}=\frac{1}{\frac{9}{11}\left(\frac{2}{11}\right)^2}-\frac{1}{\frac{9}{11}}-2=\frac{1331-44-72}{36}=\boxed{\color{blue}{\frac{135}{4}}}$$
</s>