{{ toc.name }}
{{ toc.signature }}
{{ toc.name }} {{ 'ml-btn-view-details' | message }}
{{ stepNode.name }}
Proceed to next lesson
Lesson
Exercises
Recommended
Tests
An error ocurred, try again later!
Chapter {{ article.chapter.number }}
{{ article.number }}. 

{{ article.displayTitle }}

{{ article.introSlideInfo.summary }}
{{ 'ml-btn-show-less' | message }} {{ 'ml-btn-show-more' | message }} expand_more
{{ 'ml-heading-abilities-covered' | message }}
{{ ability.description }}

{{ 'ml-heading-lesson-settings' | message }}

{{ 'ml-lesson-show-solutions' | message }}
{{ 'ml-lesson-show-hints' | message }}
{{ 'ml-lesson-number-slides' | message : article.introSlideInfo.bblockCount}}
{{ 'ml-lesson-number-exercises' | message : article.introSlideInfo.exerciseCount}}
{{ 'ml-lesson-time-estimation' | message }}

Rule

Sum of a Geometric Series

Let and be the first term and the common ratio, respectively, of a geometric sequence with terms, where The sum of the related finite geometric series or the partial sum of the infinite series can be found by using the following formula.

For an infinite series, if the common ratio is greater than and less than — in other words, if — then the sum can be found by using the following formula.

This means that the sum converges on a number. If the common ratio is less than or equal to or greater than or equal to — if — then the sum diverges. In such cases, there is no sum for the infinite geometric series.

Proof

Finite Sum
To derive the formula for a geometric series, a geometric sequence of terms with the first term and the common ratio will be considered.
All the terms will be added to express the series and will be factored out.
Then, both sides of the equation will be multiplied by and the resulting equation simplified.
Notice that the like terms are ordered in minus-plus pairs. This means that after simplifying, they will cancel out and only the first and last terms will remain.
The formula for the sum of a finite geometric series has been derived.

Proof

Infinite Sum
To derive the formula for the sum of an infinite geometric series with the standard formula for the finite geometric series will be considered.
Since is a number between and the value of becomes very small as the value of increases. In other words, it gets closer to as approaches infinity.
Therefore, can be substituted into the standard formula and the resulting equation simplified.
The formula for the sum of an infinite geometric series with has been derived.