Determine the sums of the following geometric series when they are convergent. (1)/(3^2)-(1)/(3^

step 1 In this question, we are finding the sum of an infinite geometric series. In order to do this, w

Determine the sums of the following geometric series when...

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Key Concepts

Convergence of Series

Convergence refers to the behavior of an infinite series where the partial sums approach a finite limit as more terms are added. For a geometric series, convergence is guaranteed if and only if the absolute value of the common ratio is less than one.

Geometric Series

A geometric series is a sequence of numbers where each term after the first is obtained by multiplying the previous term by a constant called the common ratio. This structure allows for a straightforward method to analyze the series and determine properties like convergence and sum.

Common Ratio

The common ratio is the fixed factor by which each term of a geometric series is multiplied to produce the next term. It is crucial for determining whether the series converges or diverges, as convergence requires the absolute value of the common ratio to be less than one.

Sum of a Convergent Geometric Series

For a convergent geometric series, the sum of the infinite series can be calculated using the formula S = a/(1 - r), where 'a' is the first term and 'r' is the common ratio. This formula simplifies the process of finding the total sum of an infinite number of terms.

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