Question

Evaluate: $\sum _{i=1}^{\infty }\frac{i^2}{2^i}$.

• A. $2$
• B. $4$
• C. $6$
• D. $28$

Answer 1

The correct option is C $6$

$S=\frac{1}{2}+\frac{4}{4}+\frac{9}{8}+\frac{16}{16}+\frac{28}{32}+...\underset{–}{-\frac{1}{2}S}=\underset{–}{\frac{1}{4}+\frac{4}{8}+\frac{9}{16}+\frac{16}{32}+...}\frac{S}{2}=\frac{1}{2}+\frac{3}{4}+\frac{5}{8}+\frac{7}{16}+\frac{9}{32}+...\underset{–}{-\frac{S}{4}=\frac{1}{4}+\frac{3}{8}+\frac{5}{16}+\frac{7}{32}+....}\frac{S}{4}=\frac{1}{2}+\frac{2}{4}+\frac{2}{8}+\frac{2}{16}+\frac{2}{32}+...$

$\frac{S}{4}=\frac{1}{2}+\frac{2/4}{1-\frac{1}{2}}=\frac{1}{2}+1=\frac{3}{2}S=6$