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Chapter 9: Problem 4

Indicate whether the given series converges or diverges. If it converges, find its sum. $$ \sum_{k=1}^{\infty}\left[5\left(\frac{1}{2}\right)^{k}-3\left(\frac{1}{7}\right)^{k+1}\right] $$

Short Answer

Expert verified
The series converges, and its sum is \(\frac{69}{14}\).

Step by step solution

01

Identify the series components

The given series is \[\sum_{k=1}^{\infty}\left[5\left(\frac{1}{2}\right)^{k}-3\left(\frac{1}{7}\right)^{k+1}\right]\]This is a combination of two geometric series:1. \[\sum_{k=1}^{\infty} 5\left(\frac{1}{2}\right)^{k}\]2. \[\sum_{k=1}^{\infty} 3\left(\frac{1}{7}\right)^{k+1}\]We will evaluate each series separately to check for convergence and sum.
02

Analyze the first series

The first series is \[\sum_{k=1}^{\infty} 5\left(\frac{1}{2}\right)^{k}\]. This represents a geometric series with first term \(a = 5 \cdot \frac{1}{2} = \frac{5}{2}\) and common ratio \(r = \frac{1}{2}\). Since \(|r| < 1\), the series converges. The sum of this geometric series is given by:\[S_1 = \frac{a}{1-r} = \frac{\frac{5}{2}}{1-\frac{1}{2}} = \frac{\frac{5}{2}}{\frac{1}{2}} = 5\]
03

Analyze the second series

The second series is \[\sum_{k=1}^{\infty} 3\left(\frac{1}{7}\right)^{k+1}\]. Rewrite the expression to start from \(k = 2\):\[\sum_{k=1}^{\infty} 3\left(\frac{1}{7}\right)^{k+1} = \sum_{k=2}^{\infty} 3\left(\frac{1}{7}\right)^{k}\]This represents a geometric series with the first term \(a = 3\left(\frac{1}{7}\right)^2 = \frac{3}{49}\) and common ratio \(r = \frac{1}{7}\). Since \(|r| < 1\), the series converges. The sum is given by:\[S_2 = \frac{a}{1-r} = \frac{\frac{3}{49}}{1-\frac{1}{7}} = \frac{\frac{3}{49}}{\frac{6}{7}} = \frac{3}{42} = \frac{1}{14}\]
04

Combine the results

Now that we have the sums of both series, we combine them to find the sum of the original series:\[S = S_1 - S_2 = 5 - \frac{1}{14} = \frac{70}{14} - \frac{1}{14} = \frac{69}{14}\]Therefore, the series converges, and its sum is \(\frac{69}{14}\).

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Key Concepts

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

Convergence
Understanding convergence is essential when dealing with infinite series. When a series converges, it means that as you keep adding more and more terms, the total sum approaches a specific finite number. In our problem, we have a series which is a combination of two separate geometric series. For a geometric series
  • The sequence is described by an initial term and a common ratio.
  • The series converges if the absolute value of the common ratio is less than 1.
This is because the powers of the common ratio shrink and become negligible as they increase, causing the series to settle at a finite sum. If the common ratio were 1 or greater, the series would continue to increase indefinitely, meaning it diverges.
Series Sum
Once a series is known to converge, we can calculate its sum. The sum of a converging geometric series is given by the formula \( S = \frac{a}{1-r} \), where \( a \) is the first term, and \( r \) is the common ratio.
  • For the first geometric series in the problem, the first term is \( \frac{5}{2} \), and the common ratio is \( \frac{1}{2} \), resulting in a sum of 5.
  • The second geometric series has a first term of \( \frac{3}{49} \) and a common ratio of \( \frac{1}{7} \), leading to a sum of \( \frac{1}{14} \).
By combining the sums of the two geometric series in the problem, the overall sum of \( \frac{69}{14} \) is obtained, confirming the convergence and quantifying the series.
Common Ratio
The common ratio is a critical element of a geometric series. It is the factor you multiply by to get from one term to the next. Recognizing the common ratio helps determine whether a series converges and aids in calculating the sum.
  • A geometric series with a common ratio \( |r| < 1 \) means each successive term gets smaller, ensuring convergence.
  • If \( |r| = 1 \), the series keeps oscillating or growing without settling, leading to divergence.
  • In our exercise, the common ratios \( \frac{1}{2} \) and \( \frac{1}{7} \) for the two series fall within the convergence threshold.
Understanding the behavior governed by the common ratio is pivotal for solving problems involving geometric series, especially when determining whether infinite terms add to a finite value.

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