**Contents:**

**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 *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 + ar ^{2} + ar^{3} + …*

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 [1]:

And this series has even terms that are negative:

The alternating geometric series can also be written in summation notation. For example [2]:

## 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 [3]:

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

[1] 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

[2] 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

[3] 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 ^{a}⁄_{1 – 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.

Watch the video for two examples, or read on below:

## 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 ^{1}⁄_{5}.

Step 2: **Confirm that the series actually converges**. The r-value for this particular series ( ^{1}⁄_{5}) 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 (^{1}⁄_{5})^{0}.

Step 4: **Set up the formula** to calculate the sum of the geometric series, ^{a}⁄_{1-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 ^{5}⁄_{4}

*That’s it!*

## What is the rth Moment?

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

The

rth moment = (x_{1}^{r}+ x_{2}^{r}+ x_{3}^{r}+… + x_{n}^{r}) / n.

**See**: Rth moments and moments defined

## 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

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