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The Infinite Geometric Series Calculator will calculate:

- The sum of all terms of an infinite geometric series when the first term and the common ratio are given.

**Infinite Geometric Series Calculator Parameters:** The common ratio is less than 1 and more than -1.

Result of the geometric series (S_{∞}) = |

Infinite Geometric Series Calculations and Formula |
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S_{∞} = y_{1}/1 - RS _{∞} = /1 - S _{∞} = /S _{∞} = |

Infinite Geometric Series Calculator Input Values |

First term of the infinite geometric series (y_{1}) = |

Common ratio of the series (R) = |

Please note that the formula for each calculation along with detailed calculations is shown further below this page. As you enter the specific factors of each infinite geometric series calculation, the Infinite Geometric Series Calculator will automatically calculate the results and update the formula elements with each element of the infinite geometric series calculation. You can then email or print this infinite geometric series calculation as required for later use.

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The Infinite Geometric Series Calculator has practical application and use in the following fields and disciplines

An infinite series is a series that has an infinite number of terms. Infinite series never terminate, i.e. they have no final value. An infinite series is either expressed in a summation notation:

S = __∞____∑___{n = 1}x_{n}

or by using three dots after the last known term to indicate the continuation to infinity, i.e.

S = x + x_{2} + x_{3} + ⋯ + x_{n} + ⋯

For example,

S = 3 + 7 + 11 + 15

is a finite series, as the last term of the series is known (15). On the other hand,

S = 3 + 7 + 11 + 15 + ⋯

is an infinite series, as it continues beyond the last known term (15). This is indicated by the three dots after this term

There are two types of infinite geometric series: **converging** series and **diverging** series.

A converging infinite series has a finite value that represents the exact sum towards which the series points, despite having an infinite number of terms. For example, the following series

S = *1**/**2* + *1**/**4* + *1**/**8* + *1**/**16* + ⋯

is both infinite and converging, as the sum points towards 1 (the whole), as shown in the corresponding figure in math tutorial 12.4. Here we will deal only with infinite converging geometric series, The condition for an infinite geometric series to be converging is that the common ratio be between -1 and 1 (-1 < R < 1).

There are some geometrical or factorisation methods that help calculate the value of infinite geometric series. However, these methods are limited and can be used only in some specific situations. The best method for calculating infinite geometric series is to use a general formula that allows finding the value of all such series, regardless of their complexity. We start from the formula of geometric progression

S_{n} = *y*_{1} ∙ (1 - R^{n})*/**1 - R*

(where y_{1} is the first term of the series and R is the common ratio) and since for n → ∞ the value of R_{n} points towards 0 (R_{n} → 0), we obtain the following formula for infinite geometric series

S_{n} = *y*_{1} ∙ 1 - 0*/**1 - R*

S_{∞} = *y*_{1}*/**1 - R*

S

In the example obtained earlier, we have y_{1} = ** 1/2** and R =

S_{∞} = *y*_{1}*/**1 - R*

=*1**/**2**/**1 - **1**/**2*

=*1**/**2**/**1**/**2*

= 1

=

=

= 1

This formula helps calculate the value of much more complicated series, such as the following one:

S_{∞} = *5**/**2* + *5**/**6* + *5**/**18* + ⋯

We have y_{1} = ** 5/2** and R =

S_{∞} = *5**/**2* + *5**/**2* ∙ *1**/**3* + *5**/**6* ∙ *1**/**3* + ⋯

Therefore, the value of this series is

S_{∞} = *y*_{1}*/**1 - R*

=*5**/**2**/**1 - **1**/**3*

=*5**/**2**/**2**/**3*

=*5**/**2* ∙ *3**/**2*

=*15**/**4*

=

=

=

=

The following Math tutorials are provided within the Sequences and Series section of our Free Math Tutorials. Each Sequences and Series tutorial includes detailed Sequences and Series formula and example of how to calculate and resolve specific Sequences and Series questions and problems. At the end of each Sequences and Series tutorial you will find Sequences and Series revision questions with a hidden answer that reveal when clicked. This allows you to learn about Sequences and Series and test your knowledge of Math by answering the revision questions on Sequences and Series.