Prove that
Question:

Prove that $\left(1-\frac{1}{3}+\frac{1}{3^{2}}-\frac{1}{3^{3}}+\frac{1}{3^{4}} \ldots \infty\right)=\frac{3}{4}$

Solution:

It is Infinite Geometric Series.

Here, a = 1,

$r=\frac{\frac{-1}{3}}{1}=\frac{-1}{3}$

Formula used: Sum of an infinite Geometric series $=\frac{a}{1-r}$

$\therefore$ Sum $=\frac{1}{1-\frac{-1}{3}}=\frac{1 \times 3}{3+1}=\frac{3}{4}=$ R.H.S.

Hence, Proved that $\left(1-\frac{1}{3}+\frac{1}{3^{2}}-\frac{1}{3^{3}}+\frac{1}{3^{4}} \ldots \infty\right)=\frac{3}{4}$