<s>[INST] Sum of series = S 
If $S=\frac35+\frac{10}{5^2}+\frac{21}{5^3}+\frac{36}{5^4}+\frac{55}{5^5}+\cdots$

In this we have to find value of S
Intianlly I thought it is geometric series .
After that I try to makes terms which can cancel other terms.
 [/INST] We have
$$\frac{S}{5} = \frac{3}{5^2}+\frac{10}{5^3}+\frac{21}{5^4}+\frac{36}{5^5}+\frac{55}{5^6}+\cdots$$
Hence
$$S_1 = \frac{4S}{5} = S - \frac{S}{5} = \frac{3}{5}+\frac{7}{5^2}+\frac{11}{5^3}+\frac{15}{5^4}+\frac{19}{5^6}+\cdots$$
$$\frac{1}{5}S_1 = \frac{3}{5^2}+\frac{7}{5^3}+\frac{11}{5^4}+\frac{15}{5^5}+\frac{19}{5^7}+\cdots$$
Hence
\begin{align*}
\frac{4}{5}S_1 &= \frac{3}{5}+\frac{4}{5^2} + \frac{4}{5^3} + \cdots \\
&=\frac{3}{5} +\frac{4}{5^2}\frac{1}{1-\frac{1}{5}}\\
&= \frac{3}{5} + \frac{1}{5} = \frac{4}{5}
\end{align*}
Hence $S_1 = 1$ and $S = \frac{5}{4}$
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