Practice Infinite Geometric Series in Math
Use these practice problems to test your method after reviewing the concept explanation and worked examples.
Quick Recap
The sum of all terms of a geometric sequence with common ratio |r| < 1. The infinite sum converges to \frac{a}{1-r}, where a is the first term.
If each term is a fixed fraction of the previous one, the terms shrink fast enough that the total sum stays finite. Imagine walking halfway to a wall, then half the remaining distance, then half again—you approach the wall but the total distance is finite (exactly the full distance to the wall).
Example 1
easyFind the sum of the infinite geometric series \displaystyle\sum_{n=0}^{\infty} \left(\frac{1}{3}\right)^n.
Example 2
mediumConvert the repeating decimal 0.\overline{27} to a fraction using an infinite geometric series.
Example 3
easyFind the sum: 4 + 2 + 1 + \frac{1}{2} + \frac{1}{4} + \cdots
Example 4
hardFor what values of x does \displaystyle\sum_{n=0}^{\infty} (x-1)^n converge, and what is the sum?