Find the sum to infinity of the geometric series whose first term is $2$ and common ratio is $\frac{1}{3}$ .

medium 3 marks Class 11 · Mathematics · Sequences and Series

Answer

3

The correct answer is $3$ — Hence the sum to infinity is $3$ .

Step-by-step solution

A geometric series converges to $\frac{a}{1 - r}$ if $∣ r ∣ < 1$ .
Here $a = 2$ , $r = \frac{1}{3}$ .
Since $∣ \frac{1}{3} ∣ < 1$ , the series converges.
Compute $\frac{2}{1 - \frac{1}{3}} = \frac{2}{\frac{2}{3}} = 2 \times \frac{3}{2} = 3$ .
Hence the sum to infinity is $3$ .

Common mistake: Using the formula

S_{n} = \frac{a ( 1 - r ^{n} )}{1 - r}

without taking the limit as

n \to \infty

Concept tested: Sum to infinity of convergent GP