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Chapter 7: Problem 71

Define a geometric series, state when it converges, and give the formula for the sum of a convergent geometric series.

Short Answer

Expert verified
A geometric series is defined as a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. It converges when the absolute value of the common ratio is less than 1. The sum of a convergent geometric series is given by \( S = \frac{a}{1 - r} \), where 'a' is the first term and 'r' is the common ratio of the series.

Step by step solution

01

Definition of Geometric Series

A geometric sequence or series is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. For example, the sequence 2, 6, 18, 54, ... is a geometric sequence with common ratio 3.
02

Convergence of Geometric Series

A geometric series converges if the absolute value of the common ratio is less than 1 i.e. \( -1 < r < 1 \) where r is the common ratio.
03

Sum of Convergent Geometric Series

For a geometric series that converges, the sum of the series 'S' is given by: \( S = \frac{a}{1 - r} \), where 'a' is the first term and 'r' is the common ratio of the series.

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Most popular questions from this chapter

Find the sum of the convergent series. $$ \sum_{n=1}^{\infty} \frac{8}{(n+1)(n+2)} $$

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. The series \(\sum_{n=1}^{\infty} \frac{n}{1000(n+1)}\) diverges.

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(a) write the repeating decimal as a geometric series and (b) write its sum as the ratio of two integers $$ 0 . \overline{81} $$
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