 # Sum of a Convergent Geometric Series

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Related article: Finite Geometric sequences.

## What is a Geometric Series?

A geometric series (or geometric progression) is one where every two successive terms have the same ratio. Once a common factor is removed from the series, you end up with a value raised to a series of consecutive powers. This type of series have important applications in many fields, including economics, computer science, and physics. An example of a gemetric series. This one decreases by a common ratio of ½.

An infinite series is the description of an operation where infinitely many quantities, one after another, are added to a given starting quantity. Any geometric series can be written as

a + ar + ar2 + ar3 + …

where a is the initial term (also called the leading term) and r is the ratio that is constant between terms. We call this ratio the common ratio.

A geometric series can either be finite or infinite.

• A finite series converges on a number. For example, 1/2 + 1/4 + 1/8…converges (i.e. settles on) on 1.
• An infinite geometric series does not converge on a number. For example, 10 + 20 + 20… does not converge (it just keeps on getting bigger).

## Finite Geometric Series

A finite geometric series has a set number of terms. For example, instead of having an infinite number of terms, it might have 10, 20, or 99. As long as there’s a set end to the series, then it’s finite. For example, all of the following are finite geometric series:

 Geometric Series Starting Term (a) Ratio (r) Number of terms (n) 2 + 4 + 8 + 16 + 32 2 2 5 2 – 20 + 200 – 2000 3 -10 4  5 ½ 101

## Infinite Geometric Series

The series: is an infinite geometric series. The “…” at the end of the series means that this particular series goes on to infinity. Each term is equal to the previous term times a constant, the common ratio. Since this common ratio is ½, we know this series converges, and we know it will approach (½)/(1 – ½) = 1 as the number of terms goes to infinity.

Another example of a this type of series is

2 + 4 + 8 + 16 + 32 + …,

Here again each term is equal to the previous term times a constant, so we know our series is geometric. The constant, 2, is greater than 1, so the series will diverge.

## Alternating Geometric Series

The alternating geometric series has terms that alternate in sign: either the odd terms are negative or the even terms are negative. For example, the following series has odd terms that are negative : And this series has even terms that are negative: The alternating geometric series can also be written in summation notation. For example : ## Convergence of an Alternating Geometric Series

An alternating geometric series will converge if its terms consistently get smaller and approach zero. Any series of this type with a small common ratio will rapidly converge. Therefore, you only need to sum a few terms.

Not all alternating geometric series will converge. To test convergence, use the alternating series test.

However, if then use the nth term test instead.

## Convergence in the Complex Plane

Alternating geometric series are either ascending or descending. The following table shows the conditions for convergence in the complex plane :

 Case Ascending Descending Divergent |z| > 1 |z| < 1 Divergent |z| = -1 |z| = -1 Oscillating |z| = 1 ≠ -z |z| = 1 ≠ -z Absolutely Convergent |z| < 1 |z| > 1 Totally Convergent |z| ≤ 1 -ε |z| ≥ 1 + δ Where 0 < ε < 1 δ > 0

## References

 Larson, R. et al. (1995). Calculus of a Single Variable: Early Transcendental Functions. Chapter 9. Infinite Series. Cengage Learning. Retrieved April 5, 2021 from: http://www.math.utep.edu/Faculty/nsharma/public_html/LarCalc10_ch09_sec5.pdf
 Hassoun, M. ECE 4330 Lecture 3 Math Review (Continued). Retrieved April 5, 2021 from: https://neuron.eng.wayne.edu/auth/ece4330/lectures/lecture_3_ece4330t.pdf
 Braga da Costa Campos, L. (2010). Complex Analysis with Applications to Flows and Fields. CRC Press.

## Sum of a Convergent Geometric Series

The sum of a convergent geometric series can be calculated with the formula a1 – r, where “a” is the first term in the series and “r” is the number getting raised to a power. A geometric series converges if the r-value (i.e. the number getting raised to a power) is between -1 and 1.
A geometric series converges if and only if the absolute value of the common ratio, |r|, is less than 1. As a formula, that’s if:
0 < | r | < 1
Where r is the common ratio.
In this case, the series will approach a / (1 – r).

If r is greater or equal to 1, the series diverges. As a formula, that’s if:
| r | ≥ 1

In general, computing the sums of series in calculus is extremely difficult and is beyond the scope of a calculus II course. However, the geometric series is an exception.

## Sum of a Convergent Geometric Series: Example

Example problem: Find the sum of the following geometric series: Step 1: Identify the r-value (the number getting raised to the power). In this sample problem, the r-value is 15.

Step 2: Confirm that the series actually converges. The r-value for this particular series ( 15) is between -1 and 1 so the series does converge.

Step 3: Find the first term. Get the first term is obtained by plugging the bottom “n” value from the summation. The bottom n-value is 0, so the first term in the series will be (15)0.

Step 4: Set up the formula to calculate the sum of the geometric series, a1-r. “a” is the first term you calculated in Step 3 and “r” is the r-value from Step 1: The sum of this particular geometric series is 54
That’s it!

## What is the rth Moment?

A “rth moment” refers to the following geometric series:

The rth moment = (x1r + x2r + x3r +… + xnr) / n.

## References

Aomoto, K. & Kita, M. (2011). Theory of Hypergeometric Functions. Springer Science and Business Media.
Berresford, G. & Rocket, A. (2015). Applied Calculus. Cengage Learning.
Callahan, J. (2010). Advanced Calculus: A Geometric View. Springer Science & Business Media.
Erdelyi, A. Ed. (1955). Higher Transcendental Functions. McGraw-Hill.
Pearson, J. et al. (2017) Numerical methods for the computation of the confluent and Gauss hypergeometric functions. Numer Algor (2017) 74:821–866
Seaborn. Hypergeometric Functions and Their Applications. Retrieved November 26, 2019 from: https://books.google.com/books?id=HJXkBwAAQBAJ

CITE THIS AS:
Stephanie Glen. "Sum of a Convergent Geometric Series" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/sequence-and-series/sum-of-a-convergent-geometric-series/
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