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Chapter 9: Problem 12

Indicate whether the given series converges or diverges. If it converges, find its sum. $$ \sum_{k=1}^{\infty} \frac{4^{k+1}}{7^{k-1}} $$

Short Answer

Expert verified
The series converges, and its sum is \(\frac{16}{3}\).

Step by step solution

01

Simplify the General Term

The given series is \[\sum_{k=1}^{\infty} \frac{4^{k+1}}{7^{k-1}}.\]First, simplify the general term by separating the powers:\[\frac{4^{k+1}}{7^{k-1}} = \frac{4 \times 4^k}{7^{k-1}} = 4 \times \frac{4^k}{7^k} \times 7.\]This simplifies to:\[\frac{4 \times 7}{7} \times \left(\frac{4}{7}\right)^k = 4 \times \left(\frac{4}{7}\right)^k.\]
02

Identify Geometric Series Components

The simplified general term \[4 \cdot \left(\frac{4}{7}\right)^k\]describes a geometric series. Compare it with \[ar^k,\]where the common ratio \(r = \frac{4}{7}\) and the first term \(a = 4 \cdot \frac{4}{7}\).
03

Check the Convergence Criterion for Geometric Series

A geometric series converges if the common ratio \(|r| < 1\). For this series, \(r = \frac{4}{7}\), and since \(\frac{4}{7} < 1\), the series converges.
04

Calculate the Sum of the Convergent Series

The sum \(S\) of a convergent geometric series with first term \(a\) and common ratio \(r\) is given by:\[S = \frac{a}{1 - r}.\]Here, \(a = 4 \cdot \frac{4}{7} = \frac{16}{7}\) and \(r = \frac{4}{7}\), so:\[S = \frac{\frac{16}{7}}{1 - \frac{4}{7}} = \frac{\frac{16}{7}}{\frac{3}{7}} = \frac{16}{3}.\]

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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 sum of terms that follow a specific pattern, where each term after the first is obtained by multiplying the previous one by a fixed number. This fixed number is called the common ratio. Geometric series are everywhere in mathematics and are valuable for solving problems involving repeated multiplication. They take the general form:
  • First term: \( a \)
  • Common ratio: \( r \)
The re-iterative structure of geometric series makes them easy to analyze and calculate. This kind of series can either terminate (having a finite number of terms) or continue indefinitely, called infinite geometric series. When dealing with convergence, infinite series are of particular interest.
Common Ratio
The common ratio in a geometric series is the constant factor between consecutive terms. It is calculated by dividing the second term by the first term. For a series to be classified as geometric, the ratio must remain constant throughout all terms.
For example, in the geometric series identified in the exercise, the common ratio \( r \) is calculated as:
  • \( r = \frac{4}{7} \)
The fixed nature of the common ratio enables us to predict the behavior of the series over time. When dealing with convergence, the magnitude of this ratio determines whether the infinite series sums up to a finite number.
Sum of Series
In a geometric series, if the series converges, it is possible to calculate its sum. For a convergent infinite geometric series with first term \( a \) and common ratio \( r \), the sum \( S \) is calculated using the formula:
  • \( S = \frac{a}{1 - r} \)
This formula is a powerful tool because it provides a straightforward way to find the sum, without having to manually add an infinite number of terms.
In the context of the exercise, once the series was identified as a geometric series, the sum was calculated as:
  • First term, \( a = \frac{16}{7} \)
  • Common ratio, \( r = \frac{4}{7} \)
  • Sum: \( S = \frac{16}{3} \)
Calculating the sum of a convergent geometric series is simple and effective with this formula.
Convergence Criterion
A crucial question when dealing with infinite series is whether they converge or diverge. For geometric series, there is a straightforward "rule of thumb" called the convergence criterion. This criterion considers the absolute value of the common ratio \(|r|\).
  • The series converges if \(|r| < 1\).
  • The series diverges if \(|r| \geq 1\).
The convergence criterion stems from the behavior of repeated multiplication, where multiplying a number less than one by itself repeatedly causes the values to shrink towards zero.
In the exercise example, since the common ratio \( \frac{4}{7} \) is less than one, the series converges. This makes it possible to determine the sum of the series using the sum formula for convergent geometric series.

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