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Sequences and Series

Finding Sums and Partial Sums of Geometric Series

Concept

Geometric Series

A geometric series is the sum of the terms of a geometric sequence.
The explicit rule of the geometric sequence above is By using this rule, the series can be written using sigma notation.
The sum of an infinite or a finite geometric series can be found by using their corresponding formulas.

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 n terms, whose first term is a1 and whose common ratio is r. 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 a1, 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 a1 is the first term, r is the common ratio, and n is the number of terms. From the given sigma notation, we can see that n=4 and r=1.5. To find a1, we can substitute n=1 into the expression
801.50
801
80
Thus, a1=80. We can substitute the noted values into the formula for
The sum of the given geometric series is 650.

Theory

Sum of an Infinite Geometric Series

When the common ratio of a geometric series is greater than 1, each term becomes larger and larger as the series continues. Similarly, when the common ratio is less than -1, each term becomes more and more negative. In these cases, and when r=±1, the infinite series has no sum. On the other hand, when r is a fraction such that
-1<r<1,
becomes very small as the number of terms, n, increases. In fact,
Thus, the standard formula for the sum of a geometric series, becomes the following for an infinite geometric series with -1<r<1.

Rule

Partial Sum of an Infinite Geometric Series

If the common ratio of an infinite geometric series is less than or equal to -1, or greater than or equal to 1, 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.

fullscreen
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
-1<r<1.
Thus, we must first determine r, which can be done by dividing the second term, a2, by a1.
Thus, the common ratio is Since r<1, it's possible to find the sum of the series. We can substitute a1 and r into the following formula.
S=1.5
The sum of the series is 1.5.
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