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This series can also be written in summation notation as k = 1 2 k , where the upper limit of summation is infinity. Because the terms are not tending to zero, the sum of the series increases without bound as we add more terms. Therefore, the sum of this infinite series is not defined. When the sum is not a real number, we say the series diverges .

Determining whether the sum of an infinite geometric series is defined

If the terms of an infinite geometric series approach 0, the sum of an infinite geometric series can be defined. The terms in this series approach 0:

1 + 0.2 + 0.04 + 0.008 + 0.0016 + ...

The common ratio r  = 0 .2 . As n gets very large, the values of r n get very small and approach 0. Each successive term affects the sum less than the preceding term. As each succeeding term gets closer to 0, the sum of the terms approaches a finite value. The terms of any infinite geometric series with 1 < r < 1 approach 0; the sum of a geometric series is defined when 1 < r < 1.

Determining whether the sum of an infinite geometric series is defined

The sum of an infinite series is defined if the series is geometric and 1 < r < 1.

Given the first several terms of an infinite series, determine if the sum of the series exists.

  1. Find the ratio of the second term to the first term.
  2. Find the ratio of the third term to the second term.
  3. Continue this process to ensure the ratio of a term to the preceding term is constant throughout. If so, the series is geometric.
  4. If a common ratio, r , was found in step 3, check to see if 1 < r < 1 . If so, the sum is defined. If not, the sum is not defined.

Determining whether the sum of an infinite series is defined

Determine whether the sum of each infinite series is defined.

  1. 12 + 8 + 4 + 
  2. 3 4 + 1 2 + 1 3 + ...
  3. k = 1 27 ( 1 3 ) k
  4. k = 1 5 k
  1. The ratio of the second term to the first is 2 3 , which is not the same as the ratio of the third term to the second, 1 2 . The series is not geometric.
  2. The ratio of the second term to the first is the same as the ratio of the third term to the second. The series is geometric with a common ratio of 2 3 . The sum of the infinite series is defined.

  3. The given formula is exponential with a base of 1 3 ; the series is geometric with a common ratio of 1 3 . The sum of the infinite series is defined.
  4. The given formula is not exponential; the series is not geometric because the terms are increasing, and so cannot yield a finite sum.
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Determine whether the sum of the infinite series is defined.

1 3 + 1 2 + 3 4 + 9 8 + ...

The sum is defined. It is geometric.

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24 + ( −12 ) + 6 + ( −3 ) + ...

The sum of the infinite series is defined.

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k = 1 15 ( 0.3 ) k

The sum of the infinite series is defined.

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Finding sums of infinite series

When the sum of an infinite geometric series exists, we can calculate the sum. The formula for the sum of an infinite series is related to the formula for the sum of the first n terms of a geometric series.

S n = a 1 ( 1 r n ) 1 r

We will examine an infinite series with r = 1 2 . What happens to r n as n increases?

( 1 2 ) 2 = 1 4 ( 1 2 ) 3 = 1 8 ( 1 2 ) 4 = 1 16

The value of r n decreases rapidly. What happens for greater values of n ?

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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