Infinite Geometric Series Formula
Before learning the infinite geometric series formula, let us recall what is a geometric series. A geometric series is the sum of a sequence wherein every successive term contains a constant ratio to its preceding term. An Infinite geometric series has an infinite number of terms and can be represented as a, ar, ar^{2}, ..., to ∞. Let us learn the infinite geometric series formula in the upcoming section.
What Is the Infinite Geometric Series Formula?
The infinite geometric series formula is used to find the sum of all the terms in the geometric series without actually calculating them individually. The infinite geometric series formula is given as:
\(S_{n}=\dfrac{a}{1r}\)
Where
 a is the first term
 r is the common ratio
Let us see the applications of the infinite geometric series formula in the following section.
Solved Examples Using Infinite Geometric Series Formula

Example 1: Find the sum of the terms 1/9 + 1/27 + 1/81 + ... to ∞?
Solution:
To find: Sum of the geometric series
Given:
a = 1/9, r = 1/3
Using the infinite geometric series formula,
S_{n} = a /(1r)
S_{n} = (1/9)( 1  1/3)
S_{n} = 1/6
Answer: The sum of the given terms is 1/6

Example 2: Calculate the sum of series 1/5, 1/10, 1/20, .... if the series contains infinite terms.
Solution:
To find: Sum of the geometric series
Given:
a = 1/5, r = 1/5
Using the infinite geometric series formula,
S_{n} = a /(1r)
S_{n} = (1/5)( 1  1/5)
S_{n} = 1/4
Answer: The sum of the given terms is 1/4