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

step 1 Does this infinite series converge or diverge? To figure this out, we're going to need to look a

Determine whether each infinite geometric series converges...

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Key Concepts

Infinite Geometric Series

An infinite geometric series is a sum of the form a + ar + ar^2 + ar^3 + ..., where a is the first term and r is the common ratio. Such series continue indefinitely, and their behavior is determined by the size of the common ratio.

Common Ratio

The common ratio is the factor by which each term in a geometric series is multiplied to produce the subsequent term. It is a key parameter that influences the convergence or divergence of the series.

Convergence Criterion

For an infinite geometric series to converge, the absolute value of the common ratio must be less than 1, specifically |r| < 1. This condition ensures that the successive terms diminish in magnitude, allowing the series to sum to a finite value.

Sum Formula for a Convergent Series

If an infinite geometric series converges, its sum can be calculated using the formula S = a / (1 - r), where a is the first term and r is the common ratio. This formula provides a concise way to find the total sum of the series when it converges.

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