Prove that (1+1/3 + 1/3^2 + 1/3^3 + 1/3^4.... ∞) = 3/4

Question

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

Answer 1

It is Infinite Geometric Series. 

Here, a = 1,

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   \(\left(1 + \frac{1}{3} + \frac{1}{3^2} + \frac{1}{3^3} + \frac{1}{3^4}..... \infty\right)\) = \(\frac{3}{4}\)