Question

Find the sum of the convergent series. $$ 1+0.1+0.01+0.001+\cdots $$

Short answer

The sum of the series is 1.1111

Step-by-step solution

Identify the first term and the common ratio

The first term \(a\) is 1 and the common ratio \(r\) is 0.1

Apply the formula for the sum of a convergent geometric series

Using the formula \(S = \frac{a}{1-r}\), we substitute \(a = 1\) and \(r = 0.1\) into the formula. Thus the sum \(S\) becomes \(S = \frac{1}{1-0.1} = \frac{1}{0.9}\)

Simplify the expression

By simplifying the expression we get the sum of the series as \(S = \frac{1}{0.9} = 1.1111\)