Determine whether each infinite geometric series converges or diverges. If it converges, find it

step 1 An infinite geometric sequence is defined as summation running from k equal to 1 to infinity, 5

Determine whether each infinite geometric series converges...

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Key Concepts

Sum Formula for Convergent Series

When an infinite geometric series converges, its sum can be calculated using the formula S = a / (1 - r), where a is the first term of the series, and r is the common ratio. This formula is derived from the limit of the partial sums of the series as the number of terms approaches infinity.

Infinite Geometric Series

An infinite geometric series is a sum of an infinite number of terms where each term is produced by multiplying the previous term by a constant factor known as the common ratio. These series are often written in the form a + ar + ar^2 + ar^3 + ... where a is the first term and r is the common ratio.

Convergence Criteria

The convergence of an infinite geometric series depends on the value of the common ratio r. Specifically, the series converges if the absolute value of r is less than 1 (|r| < 1), and it diverges if |r| ? 1. This condition ensures that successive terms become progressively smaller, allowing the series to approach a finite limit.

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