Series and Their Notations: Learn It 4

Infinite Geometric Series

Thus far, we have looked only at finite series. Sometimes, however, we are interested in the sum of the terms of an infinite sequence rather than the sum of only the first n terms. An infinite series is the sum of the terms of an infinite sequence. An example of an infinite series is [latex]2+4+6+8+\dots[/latex].

This series can also be written in summation notation as [latex]\sum\limits _{k=1}^{\infty} 2k[/latex], 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.

If the terms of an infinite geometric series approach [latex]0[/latex], the sum of an infinite geometric series can be defined. As [latex]n[/latex] gets large, the values of of [latex]r^n[/latex] get very small and approach [latex]0[/latex]. Each successive term affects the sum less than the preceding term. As each succeeding term gets closer to [latex]0[/latex], the sum of the terms approaches a finite value. The terms of any infinite geometric series with [latex]-1 < r < 1[/latex] approach [latex]0[/latex]; the sum of a geometric series is defined when [latex]-1 < r < 1[/latex].

determining whether the sum of an infinite geometric series is defined

If the terms of an infinite geometric series approach [latex]0[/latex], the sum of an infinite geometric series can be defined.

 

The sum of an infinite series is defined if the series is geometric and [latex]-1 < r < 1[/latex].

The terms in this series approach [latex]0[/latex]:

[latex]1+0.2+0.04+0.008+0.0016+\dots[/latex]

The common ratio is [latex]r=0.2[/latex].

How To: 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, [latex]r[/latex], was found in step 3, check to see if [latex]-1 < r < 1[/latex]. If so, the sum is defined. If not, the sum is not defined.
Determine whether the sum of each infinite series is defined.

  1. [latex]12+8+4+\dots[/latex]
  2. [latex]\dfrac{3}{4}+\dfrac{1}{2}+\dfrac{1}{3}+\dots[/latex]
  3. [latex]\sum\limits _{k=1}^{\infty}{27}\cdot\left(\dfrac{1}{3}\right)^k[/latex]
  4. [latex]\sum\limits _{k=1}^{\infty}{5k}[/latex]