Question

The value of the summation $\sum _{i=1}^{\infty }\frac{2i-1}{2^{i+1}}$ is

• A. $\frac{3}{2}$
• B. $\frac{4}{3}$
• C. $\frac{5}{3}$
• D. $3$

Answer 1

The correct option is A $\frac{3}{2}$

Let $S=\sum _{i=1}^{\infty }\frac{2i-1}{2^{i+1}}$
$\Rightarrow S=\frac{1}{4}+\frac{3}{8}+\frac{5}{16}+\frac{7}{32}+\dots$
$\Rightarrow 4S=(1+\frac{3}{2}+\frac{5}{4}+\frac{7}{8}+\dots )$
Now clearly, the series is an AGP with $a=1,d=2,r=\frac{1}{2}$.
Hence, using the formula for the sum of infinite terms of AGP, we get
$S=\frac{1}{4}(\frac{1}{1-\frac{1}{2}}+\frac{2\cdot \frac{1}{2}}{{(1-\frac{1}{2})}^2})$
$=\frac{1}{4}(2+4)=\frac{3}{2}$