Problem 71
Question
Define a geometric series, state when it converges, and give the formula for the sum of a convergent geometric series.
Step-by-Step Solution
Verified Answer
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.
1Step 1: 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.
2Step 2: 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.
3Step 3: 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.
Other exercises in this chapter
Problem 70
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