Find the sum of each infinite geometric series, if it exists. (1)/(8)+(1)/(32)+(1)/(128)+⋯

Name: Problem 41, Chapter 11: Algebra 2
Uploaded: 2020-04-01T05:06:28-08:00
Duration: 2 min 8 s
Channel: John Connell
Description: Find the sum of each infinite geometric series, if it exists. (1)/(8)+(1)/(32)+(1)/(128)+⋯

step 1 So we want to find the sum of this infinite geometric series, the one that starts with 1 8th and

Find the sum of each infinite geometric series, if it...

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Key Concepts

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 factor called the common ratio. This concept is fundamental in understanding patterns that exhibit exponential growth or decay, and it forms the basis for examining series sums, convergence, and applications across various branches of mathematics.

Convergence of Infinite Series

In the context of infinite series, convergence refers to the condition where the sum of all terms approaches a fixed finite value as more terms are added. For geometric series, convergence occurs if the absolute value of the common ratio is less than one. This criterion is crucial for determining whether the infinite sum exists and has a meaningful, finite limit.

Formula for the Sum of an Infinite Geometric Series

The sum of an infinite geometric series that converges 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 by incorporating the convergence condition and eliminating the need to manually sum an infinite number of terms.

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