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Finding Sums and Partial Sums of Geometric Series

Concept

Geometric Series

When the terms in a geometric sequence are written as a sum of terms rather than a list of numbers, it is called a geometric series. If the sequence is short enough, such as writing and calculating the sum is straightforward. Sigma notation can also be used. For the series written above, the explicit rule of the related geometric sequence is Thus, the series can be written as If the series had been infinite, the infinity symbol would be above the sigma, instead of When a series is finite, its sum can be found using the formula for a sum of a geometric series.

It can also be used when calculating partial sums of an infinite series.
Proof

Sum of a Geometric Series

What follows is a derivation of the formula for the sum of a geometric series.
Consider a geometric series of terms, whose first term is and whose common ratio is The sum of the series can be written as Notice that the series is a polynomial, and can be written in standard form. Since all of the terms contain a factor of it can be factored out of the right-hand side. Since the sum above is a polynomial, polynomial division can be used to rewrite it. In fact, one polynomial identity gives Thus, the formula for the sum of a geometric series can be written as follows.


Q.E.D.
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Exercise

Calculate the sum of the geometric series.

Show Solution
Solution
To find the sum, we can use the formula for a geometric series where is the first term, is the common ratio, and is the number of terms. From the given sigma notation, we can see that and To find we can substitute into the expression
Thus, We can substitute the noted values into the formula for
The sum of the given geometric series is
Theory

Sum of an Infinite Geometric Series

When the common ratio of a geometric series is greater than each term becomes larger and larger as the series continues. Similarly, when the common ratio is less than each term becomes more and more negative. In these cases, and when the infinite series has no sum. On the other hand, when is a fraction such that becomes very small as the number of terms, increases. In fact, Thus, the standard formula for the sum of a geometric series, becomes the following for an infinite geometric series with

Rule

Partial Sum of an Infinite Geometric Series

If the common ratio of an infinite geometric series is less than or equal to or greater than or equal to the sum of the series does not exist. However, it's possible to find a partial sum, or the sum of the first several terms in the series. The partial series can be thought of as a finite series. Thus, its sum can be found using the formula for a finite geometric series.

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Exercise

Find the sum of the geometric series.

Show Solution
Solution
To begin, recall that the sum of an infinite geometric series can only be found if Thus, we must first determine which can be done by dividing the second term, by Thus, the common ratio is Since it's possible to find the sum of the series. We can substitute and into the following formula.
The sum of the series is
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