Divergent Sequences and Series
A divergent sequence is a sequence whose terms do not have a finite limit. This means that as the sequences continues, the value of the terms will tend toward positive or negative infinity. Take for example the sequence As we continue to list terms in this sequence, we note that there will never be a definite limit to what the term might be, as the value of the terms become infinitely large. This is also true when the common ratio is less than –1. The terms will alternate between positive and negative, but the absolute magnitude of the terms will head towards the infinities.
When it comes to summing an infinite divergent sequence, we find that there is not a concrete value to the sum, due to the fact that the terms themselves have no limit to their value. We refer to the sum of a sequence as a series. So if we have a divergent sequence, we also have a divergent series, since the sum also tends towards either positive or negative infinity.
Therefore, when we talk about the sum of infinite geometric sequences, we are working primarily with convergent sequences, or sequences with a common ration between –1 and 1.
Because the terms in divergent sequences tend toward positive or negative infinity, the series is also divergent. For this reason, the formula for the sum of an infinite geometric sequence, which we will explore below, applies only to convergent sequences, where the common ratio is between –1 and 1.
Convergent Sequences and Series
Unlike divergent sequences, the terms in a convergent sequence tend toward a specific value. Since the common ratio in convergent sequences is between –1 and 1, as we continue to list terms in the sequence, the value actually tends towards zero. When it comes to summing the terms in a convergent sequence, since we will eventually be adding virtually zero, this also means that the series converges to a specific value as well.
The Formula for the Sum of an Infinite Geometric Sequence
You may be familiar with the formula for the sum of a finite geometric sequence, when the number of terms in the sequence is clearly defined, rather than never-ending:
However, when we are dealing with infinite sequences, we need to consider how this affects the formula when n is infinitely large. Keep in mind that with convergent sequences r is a number between –1 and 1. Applying an ever-increasing exponent to any common ratio within this range leads to a number that is getting closer and closer to zero.
In terms of the formula, this means that the numerator in the fraction actually simplifies to 1. So our formula for the sum of an infinite geometric sequence is simpler:
Using the Formula
Find the sum of the following infinite geometric sequence:
Our first task is to find the common ratio of the sequence. To do so, we can take two consecutive terms in the sequence and divide one by the other. It is important to divide the second number by the first, in order for the quotient to describe what must be multiplied by each term in the sequence to get the value of the following term:
We'll use 58.96 and 50.11. We divide 50.11 by 58.96 to get 0.85 as the common ratio.
This means that the series will converge to 640. The sum will never exceed this value.