S-2

\(\displaystyle 2).\;\; \sum_{n=1}^{\infty} \left(\frac{1}{2}\right)^n\)

\(\text{so above series can be written as}\)

\(\displaystyle \sum_{n=1}^{\infty} \left(\frac{1}{2}\right)^n = \frac{1}{2} +\frac{1}{2^2}+\frac{1}{2^3}+ \cdots \)

\(\text{as we can see this is a G.P with common ratio}\;\;\displaystyle \frac{1}{2}\)
\(\text{and first element}\;\;\displaystyle \frac{1}{2}\)

\(\text{Sum of Infinte G.P with common ratio <1 is }\)

\(\displaystyle \textbf{sum}=\frac{a}{1-r} \cdots \cdots (1)\)

\(\text{so now from the original series we have}\)

\(\displaystyle a=\frac{1}{2}\) and \(\displaystyle r=\frac{1}{2}\)

\(\text{Using \((1)\) the Sum of given infinte series is}\)

\(\displaystyle \textbf{Sum}= \frac{\frac{1}{2}}{1-\frac{1}{2}}\)

\(\text{This will give us}\)

\(\displaystyle \textbf{Sum}= 1\)

\(\displaystyle \sum_{n=1}^{\infty} \left(\frac{1}{2}\right)^n=1\)