Worked exercise — Sum of the geometric series with ratio 1/3

Problem

Compute the sum of the geometric series $\sum_{n=0}^\infty \left(\frac{1}{3}\right)^n.$

Solution

It is a geometric series with first term $a_0 = 1$ and ratio $q = 1/3$, with $|q| < 1$; it therefore converges to the sum $\sum_{n=0}^\infty q^n = \frac{1}{1-q} = \frac{1}{1-\frac13} = \frac{1}{\frac23} = \boxed{\frac{3}{2}}.$

Links

Topics: Sequences
Concepts: Geometric progression · Geometric series
Skills: Computing limits · Using formulae
Exercise type: Computing a limit