Find the sum to infinity of the following series. 0.5 + 0.05 + 0.005 + 0.0005 + .....

Analyze the question

The series 0.5 + 0.05 + 0.005 + 0.0005 + ... is a geometric progression (GP), and we need its sum to infinity.

Concept

For a GP with first term a and common ratio r, where |r| < 1:

S∞ = a / (1 − r)

Solution

• First term: a = 0.5
• Common ratio: r = 0.05 / 0.5 = 0.1 (check: 0.005 / 0.05 = 0.1)
• Since |0.1| < 1, the sum to infinity exists.

S∞ = 0.5 / (1 − 0.1) = 0.5 / 0.9 = 5/9

Check: 0.5 + 0.05 + 0.005 + ... = 0.5555... = 0.5̇, which is 5/9. ✔

Answer: The sum is 5/9, which is option D.

Exam tip: A recurring decimal like 0.555... is just a GP with r = 0.1, so a single repeated digit d gives d/9. For example, 0.333... = 3/9 = 1/3. Remember S∞ = a/(1 − r) and check that |r| < 1 first.