Series 4

$\displaystyle \sum_{i=1}^{\infty}\frac{i^{2}}{2^{i}}=m$ $\sum_{i=1}^{\infty}\frac{i^{2}}{2^{i}}=S_1=\frac{1^{2}}{2^{1}}+\frac{2^{2}}{2^{2}}+\frac{3^{2}}{2^{3}}+\frac{4^{2}}{2^{4}}...~~(1)$ $~~~~~~~~~~~\frac{{S_1}}{2}=\frac{1^{2}}{2^{2}}+\frac{2^{2}}{2^{3}}+\frac{3^{2}}{2^{4}}+\frac{4^{2}}{2^{5}}...~~(2)$ $~~~~~~~~~~~~~~~~~\rightarrow\frac{S_1}{2}=\frac{1}{2^{1}}+\frac{3}{2^{2}}+\frac{5}{2^{3}}+\frac{7}{2^{4}}...~~(1)-(2)$ $~~~~~~~\rightarrow\frac{S_1}{4}=\frac{1}{2^{2}}+\frac{3}{2^{3}}+\frac{5}{2^{4}}+\frac{7}{2^{5}}...~~(3)$ $~~~~~\rightarrow\frac{S_1}{4}=\frac{1}{2^{1}}+2(\frac{1}{2^{2}}+\frac{1}{2^{3}}+\frac{1}{2^{4}}...)~~(1)-(2)-(3)$ $~~~~~\rightarrow\frac{S_1}{4}=\frac{1}{2^{1}}+1$ $\rightarrow\color{#3D99F6}{\boxed{S_1=6}}$ $\displaystyle\frac{\displaystyle \sum_{i = 1}^{36}i}{\displaystyle \sum_{i = 1}^{36}i^3}=\frac{\frac{(36)(37)}{2}}{\frac{(36^2)(37^2)}{4}}=\frac{1}{(18)(37)}=\frac{R}{Z}$ $\Large\rightarrow\color{#3D99F6}{\boxed{m^{2}+R+Z=36+1+(18)(37)=703}}$