**Contents:**

**Related article**: Finite Geometric sequences.

## What is a Geometric Series?

A **geometric series** 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.

## 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

Berresford, G. & Rocket, A. (2015). Applied Calculus. Cengage Learning.

Callahan, J. (2010). Advanced Calculus: A Geometric View. Springer Science & Business Media.

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