Find the value of this series what is the value of this series $$\sum_{n=1}^\infty \frac{n^2}{2^n} = \frac{1}{2}+\frac{4}{4}+\frac{9}{8}+\frac{16}{16}+\frac{25}{32}+\cdots$$
I really tried, but I couldn't, help guys?

Hint:
$$\sum_{n=1}^{\infty} \frac{n^2}{2^n} = \sum_{i=1}^{\infty} (2i - 1) \sum_{j=i}^{\infty} \frac{1}{2^j}.$$
Start by using the geometric series formula on $\displaystyle \sum_{j=1}^{\infty} \frac{1}{2^j}$ to simplify the double series into a singular series. Then you will have a series that looks like $\displaystyle \sum_{i=1}^{\infty} \frac{i}{2^i}$. Just as I broke your initial series with quadratic term $n^2$ into a double series with linear term $i$, you can break this series with linear term $i$ into a double series with constant term $c$.
In a clearer form:
\begin{align}
&\frac{1}{2} + &\frac{4}{4} + &\frac{9}{8} + &\frac{16}{16} +  \dots =\\
&\frac{1}{2} + &\frac{1}{4} + &\frac{1}{8} + &\frac{1}{16} + \dots +\\
& &\frac{3}{4} + &\frac{3}{8} + &\frac{3}{16} + \dots + \\
& & & \frac{5}{8} + &\frac{5}{16} + \dots + \\
& & & &\frac{7}{16} + \dots + \\
\end{align}
Notice that each of the sums are geometric series, which can be evaluated easily.
Sorry about the weird formatting, I don't know where these spaces are coming from.

