Problem 40 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Determine whether the series converges or diverges. In this set of problems knowledge of all the convergence tests from the chapter is assumed. \(\sum_{k=1}^{\infty}\left(1+\frac{1}{k}\right)^{k}\)

Step-by-Step Solution

Answer

The series \(\sum_{k=1}^{\infty}\left(1+\frac{1}{k}\right)^{k}\) is divergent.

Step 1: Recognizing the type of problem

The problem gives an infinite series and asks to determine if it converges or diverges. The series belongs to the type \(a_n^n\), which is optimal for application of the Root Test.

Step 2: Apply the Root Test

The Root Test states that if we have a series \(\sum a_n\) and we can find \(\lim_{{k\to\infty}} |a_k|^{\frac{1}{k}} = L\), then if \(L<1\), the series converges, if \(L>1\), the series diverges, and if \(L=1\), the test is inconclusive. Apply this to our series, we get \(\lim_{{k\to\infty}} \left|\left(1+\frac{1}{k}\right)^{k}\right|^{\frac{1}{k}} = \lim_{{k\to\infty}} \left(1+\frac{1}{k}\right)\). As \(k\) approaches infinity, \(\frac{1}{k}\) approaches 0, so \(\lim_{{k\to\infty}} \left(1+\frac{1}{k}\right) =1\). This makes this test inconclusive here.

Step 3: Apply the Ratio Test

Since the Root Test did not give a clear result, let's attempt the Ratio Test. The Ratio Test states that if we can find \(\lim_{{k\to\infty}} \frac{|a_{k+1}|}{|a_k|} = L\), if \(L<1\), the series converges, if \(L>1\), the series diverges, and if \(L=1\), the test is inconclusive. Apply the Ratio Test to our series, the limit is also 1. Thus, the Ratio Test is also inconclusive and we don't have a clear determination of the series.

Step 4: Refer to the Limit of sequence

If all our traditional tests fail, we can look at the limit of the sequence itself. If the limit as n goes to infinity of \(a_n\) is not zero, the series must diverge. Here,\(\lim_{k\to\infty} \left(1+\frac{1}{k}\right)^{k}\) is a well-known limit that equals \(e\) which is approximately 2.7, and is definitely not zero. So by this, we conclude that the series is divergent.

Key Concepts

Root Test

Understanding the Root Test can be vital in determining the behavior of an infinite series. This test is particularly useful for series where each term is raised to a power, like in the provided problem. The Root Test says you calculate the limit of the absolute value of the series' terms raised to the power of \({1/k}\): \[\lim_{{k\to\infty}} |a_k|^{\frac{1}{k}} = L\].

If \(L < 1\), the series converges. If \(L > 1\), it diverges. And if \(L = 1\), the test is inconclusive. In our original problem, \(L\) equaled 1, meaning the Root Test did not provide a definitive answer. It's always essential to explore further if the test results in an inconclusive state.

Ratio Test

The Ratio Test is another method that helps us investigate the convergence of series. Here, the idea is to examine the behavior of the ratio of successive terms in a series. You find \({\lim_{{k\to\infty}} \frac{|a_{k+1}|}{|a_k|} = L}\).

If \(L < 1\), the series converges absolutely, while if \(L > 1\), it diverges. When \(L = 1\), much like the Root Test, it is inconclusive.

• Apply it when Root Test doesn't give results.
• Aids in understanding series with factorials or exponential forms.

The application of the Ratio Test in the given problem also resulted in an \(L\) of 1, prompting the necessity for alternative strategies.

Divergent Series

A divergent series is one where the sum of its terms does not settle into a fixed finite number. In simpler terms, as you add more and more terms, the sum grows without bound or oscillates indefinitely.

This can occur when the terms of a series do not tend to zero as you go farther and farther to infinity. Even well-known limits like \( \left(1+\frac{1}{k}\right)^{k} \), which approaches the number \(e\), lead to divergent series since they do not decrease to zero. Recognizing a divergent series can help you understand whether any sum might be fruitless to seek.

Limit of a Sequence

The limit of a sequence is an important concept that aids in evaluating series. A series requires the limit of its terms, as the index grows, to be zero for convergence.

In the highlighted problem, evaluating \({\lim_{k\to\infty} \left(1+\frac{1}{k}\right)^{k}}\) leads to the famously irrational number \(e\), which is clearly not zero.

• When a series’ term approaches a non-zero limit, the series diverges.
• Even interesting limits like \(e\), approximately 2.7, still cause divergence in contexts like this.

Understanding the limits of sequences helps consolidate the understanding of why a series like presented in our problem is divergent.