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# How to find the sum of an infinite geometric series?

**Solution:**

Infinite geometric series is a series that has an infinite number of terms in a certain geometric pattern.

Go through the step-by-step solution to understand the formulation of the ultimate value.

The n^{th} term of any geometric progression = ar^{(n-1)}

Here, a = 1^{st} term, r = common multiple, n = common difference

For n tending to infinity, the formula for the sum of the series is given below.

General term of the series = ar^{(n - 1)} = T_{n}

Therefore, the series can be written as - a, ar, ar_{2}, ar_{3}, ar_{4}, ... up to infinite terms.

S_{n} = a + ar + ar_{2} + ar_{3} + ar_{4} + ... up to infinite terms.

S_{n} = a(1 + r + r_{2} + r_{3} + ...)

S_{n} = a/(1 - r) {Sum of an infinite geometric progression with common difference r is 1 / (1 - r)

Thus, sum of the infinite geometric progression can be solved using the formula, a(1 / 1 - r).

## How to find the sum of an infinite geometric series?

**Summary:**

The sum of the infinite geometric progression can be solved using the formula, a(1 / 1 - r), where a is the first term and r is the common ratio.

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