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Convergent Infinite Geometric Series
An infinite geometric series is convergent when the absolute value of its common ratio satisfies . In a convergent series, each successive term is smaller than the one before it, so the partial sums grow more and more slowly and approach a fixed limiting value rather than increasing without bound.
For example, the series has and . Since , the series is convergent. Computing a few partial sums confirms this: , , and . Each partial sum is closer to than the one before it. Because , the expression shrinks toward zero as grows, and a finite sum can be determined using the formula for the sum of an infinite geometric series.
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Ch.12 Sequences, Series and Binomial Theorem - Intermediate Algebra @ OpenStax
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Defining Infinite Series in Financial Forecasting
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