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Chapter 9: Problem 6

Indicate whether the given series converges or diverges. If it converges, find its sum. $$ \sum_{k=1}^{\infty}\left(\frac{9}{8}\right)^{k} $$

Short Answer

Expert verified
The series diverges because the common ratio is greater than 1.

Step by step solution

01

Recognize the Series Type

The series given is a geometric series of the form \( \sum_{k=1}^{\infty} ar^k \) where \( a \) is the first term and \( r \) is the common ratio. For this series, \( a = \frac{9}{8} \) and \( r = \frac{9}{8} \).
02

Determine Convergence

A geometric series converges if the absolute value of the common ratio \( r \) is less than one, \( |r| < 1 \). Here, \( r = \frac{9}{8} \), and since \( \frac{9}{8} > 1 \), the series diverges.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Geometric Series
A geometric series is a fascinating type of series where each term is a constant multiple of the previous term. This constant multiple is known as the common ratio. The structure of a geometric series can be represented as \( \sum_{k=1}^{\infty} ar^k \), where \( a \) is the first term and \( r \) is the common ratio.

Geometric series can be quickly identified by this repetition of multiplying by the same value. For example, in our exercise, the series \( \sum_{k=1}^{\infty} \left(\frac{9}{8}\right)^{k} \) clearly fits the pattern of a geometric series with an initial term \( a = \frac{9}{8} \) and a repeating multiplier (common ratio) \( r = \frac{9}{8} \).
  • The magic of geometric series is that their sum can be found under certain circumstances.
  • Specifically, when the absolute value of the common ratio is less than 1, the series converges, and the sum can be calculated with a simple formula.
Divergence
Divergence refers to a series that does not have a finite sum. In other words, as you keep adding the terms of the series, the total grows indefinitely and does not settle to a single number.

For a geometric series, divergence is determined by the common ratio \( r \). If \(|r| \geq 1\), then the series will diverge. The given exercise presents a scenario with \( r = \frac{9}{8} \), which is indeed greater than 1, confirming that it diverges.
  • When the terms of the series grow in such a way, the overall sum is not just a large number but an infinite one.
  • This property of divergence highlights the importance of the common ratio in determining the behavior of a geometric series.
Common Ratio
The common ratio is the heartbeat of a geometric series. It's what determines whether the series converges to a sum or diverges to infinity. This ratio, noted as \( r \), is the factor by which each term of the series is multiplied to get the next term.

In our exercise's series \( \sum_{k=1}^{\infty} \left(\frac{9}{8}\right)^{k} \), the common ratio is \( \frac{9}{8} \). As a result:
  • If \( |r| < 1 \), the series converges.
  • If \( |r| \geq 1 \), like in this exercise where \( r = \frac{9}{8} \), the series diverges.
Understanding the common ratio provides a quick insight into the nature of the series and is a fundamental aspect in analyzing whether a geometric series will have a finite sum.

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