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What is the formula for the sum of the infinite geometric series?

Published in Series Summation 3 mins read

The formula for the sum of an infinite geometric series is a/(1-r), where 'a' is the first term of the series and 'r' is the common ratio.

Understanding Infinite Geometric Series

An infinite geometric series is a series where each term is found by multiplying the previous term by a constant value called the common ratio. This process continues infinitely. The formula provided allows us to find a finite sum for such a series, provided the absolute value of the common ratio, |r|, is less than 1. If |r| is greater than or equal to 1, the series will not converge to a finite sum.

Formula Breakdown

Let's break down the formula:

  • a: Represents the first term in the geometric series.
  • r: Represents the common ratio, which is the value multiplied by each term to get the next term.
  • a/(1-r): This expression calculates the sum of the infinite geometric series only when the absolute value of 'r' is less than 1 (|r| < 1). If the absolute value of 'r' is greater than or equal to 1, the series diverges and does not have a finite sum.

Practical Application

Here's how to apply the formula:

  1. Identify 'a': Determine the value of the first term in the series.
  2. Calculate 'r': Find the common ratio by dividing any term by the term that precedes it.
  3. Verify convergence: Ensure that |r| < 1. If this condition is not met, the series does not converge, and the formula cannot be applied.
  4. Plug into the formula: Once convergence is verified, plug the values of 'a' and 'r' into the formula: a/(1-r).

Example

Let's consider an example: 1 + 1/2 + 1/4 + 1/8 + ....

  • The first term (a) is 1.
  • The common ratio (r) is 1/2 (each term is half of the previous term).
  • Since |1/2| < 1, the series converges.
  • The sum is then 1 / (1 - 1/2) = 1 / (1/2) = 2.

Key points

  • The formula is applicable only for converging infinite geometric series where the absolute value of the common ratio is less than 1.
  • If |r| ≥ 1, the sum does not exist, meaning the series diverges.
  • The formula provides a powerful tool for dealing with situations where infinite series have a finite sum.

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