Infinite Geometric Series Math Example 2

Follow the full solution, then compare it with the other examples linked below.

Example 2

medium
Convert the repeating decimal 0.27‾0.\overline{27} to a fraction using an infinite geometric series.

Solution

  1. 1
    0.27‾=0.272727…=27100+2710000+271000000+⋯0.\overline{27} = 0.272727\ldots = \frac{27}{100} + \frac{27}{10000} + \frac{27}{1000000} + \cdots
  2. 2
    This is a geometric series with a=27100a = \frac{27}{100} and r=1100r = \frac{1}{100}.
  3. 3
    ∣r∣=0.01<1|r| = 0.01 < 1, so S=a1−r=271001−1100=2710099100=2799=311S = \frac{a}{1-r} = \frac{\frac{27}{100}}{1 - \frac{1}{100}} = \frac{\frac{27}{100}}{\frac{99}{100}} = \frac{27}{99} = \frac{3}{11}.

Answer

0.27‾=3110.\overline{27} = \frac{3}{11}
Every repeating decimal is a geometric series in disguise. The repeating block becomes the numerator aa, and the denominator of rr is a power of 10 equal to the block length.

About Infinite Geometric Series

The sum of all terms of a geometric sequence with common ratio ∣r∣<1|r| < 1. The infinite sum converges to a1−r\frac{a}{1-r}, where aa is the first term.

Learn more about Infinite Geometric Series →

More Infinite Geometric Series Examples

Example 1 easy

Find the sum of the infinite geometric series [formula].

Example 3 medium

Find the sum of the infinite series [formula]

Example 4 easy

Find the sum: [formula]

Example 5 hard

For what values of [formula] does [formula] converge, and what is the sum?

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