Question

Find the sum, if it exists. $$200+100+50+25+12.5+\cdots$$

   Find the sum, if it exists.
$$200+100+50+25+12.5+\cdots$$

Applied Calculus

Applied Calculus

Deborah… 5th Edition

Chapter 10, Problem 19

Instant Answer

Solved by Expert Vikash Ranjan

04/07/2020

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In this case, it is a geometric series because each term is half of the previous term. Show more…

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Find the sum, if it exists. $$200+100+50+25+12.5+\cdots$$

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Key Concepts

Geometric Series

A geometric series is a sequence in which each term is produced by multiply the previous term by a constant known as the common ratio. This pattern facilitates studying the series' properties and summing it under specific criteria.

Convergence of Infinite Series

An infinite series converges if its sequence of partial sums approaches a fixed value as more terms are added. For a geometric series, convergence specifically requires the absolute value of the common ratio to be less than one, ensuring that the contributions of later terms diminish.

Sum of an Infinite Geometric Series

When an infinite geometric series converges, its total 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 effectively captures how, under the right conditions, the series approaches a finite sum despite having infinitely many terms.

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