Question

What is the sum of the infinite series \( S = \sum_{n=0}^{\infty} \frac{1}{3^n} \)?

• A. \( \frac{1}{2} \)
• B. \( \frac{3}{2} \)
• C. 2
• D. 3

Hint:

For an infinite geometric series with \( |r|<1 \), the sum is given by \( S = \frac{a}{1 - r} \).

Answer 1

The Correct Option is B

The provided series is a geometric series with an initial term of \( a = 1 \) and a common ratio of \( r = \frac{1}{3} \). The formula for the sum of an infinite geometric series is \( S = \frac{a}{1 - r} \). Substituting \( a = 1 \) and \( r = \frac{1}{3} \) into the formula yields: \[ S = \frac{1}{1 - \frac{1}{3}} = \frac{1}{\frac{2}{3}} = \frac{3}{2}. \] Therefore, the sum of this infinite geometric series is \( \frac{3}{2} \).