Question

Determine the convergence of the given series. State the test used; more than one test may be appropriate. $$\sum_{n=1}^{\infty}\left(\frac{n+2}{n+1}\right)^{n}$$

Short answer

The series diverges by the root test.

Step-by-step solution

Analyze the General Term

First, analyze the general term of the series: \( a_n = \left(\frac{n+2}{n+1}\right)^n \). As \( n \to \infty \), observe the behaviour of \( a_n \).

Apply the Limit Comparison Test

To compare \( a_n \) with known convergent or divergent series, compute the behavior of the term \( \left(1 + \frac{1}{n+1}\right)^n \) as \( n \to \infty \).

Simplify the General Term

Express \( \left(\frac{n+2}{n+1}\right)^n \) using change of variables or by rewriting: \( \left(1 + \frac{1}{n+1}\right)^n = e^{1/n+1} \approx e \) as \( n \to \infty \).

Conclude Using the Root Test

Apply the root test: compute \( L = \lim_{n \to \infty} \sqrt[n]{\left(\frac{n+2}{n+1}\right)^n} \). Simplify \( L \) to find that it approaches \( e \).

Interpret the Root Test Result

Since \( L = e > 1 \), by the root test, the series \( \sum_{n=1}^{\infty}\left(\frac{n+2}{n+1}\right)^{n} \) diverges.

Key concepts

Limit Comparison Test

The Limit Comparison Test is a powerful tool used to determine the convergence or divergence of a series by comparing it with another series whose convergence behavior is known. This test is particularly useful when dealing with complex series that resemble a simpler, well-known series.

To use the Limit Comparison Test, you need two series: the series you are analyzing, say \( a_n \), and a known comparison series \( b_n \). You calculate the limit:

• \( \ L = \lim_{n \to \infty} \frac{a_n}{b_n} \ \)

If \( L \) is a positive finite number, both series will either converge or diverge together. In the problem, the general term is simplified, and the test can help compare it to a known series such as the geometric series. However, here, further investigation through other tests is performed to confirm divergence.

Root Test

The Root Test, also known as the Cauchy root test, is another reliable method for determining the convergence of a series. This test focuses on looking at the behavior of the n-th root of the terms of a series. This can be particularly insightful when series involve powers and roots inherently.

For this test, consider a series with terms \( a_n \). You compute the quantity:

• \( \ L = \lim_{n \to \infty} \ \sqrt[n]{|a_n|} \ \)

Applying this to \( \left(\frac{n+2}{n+1}\right)^n \), we find:

• \( L = e > 1 \ \), indicating a divergent series.

The Root Test states:

• If \( L < 1 \), the series converges.
• If \( L > 1 \), the series diverges.
• If \( L = 1 \), the test is inconclusive.

Thus, in this problem, the series diverges since \( L = e \).

Divergent Series

A divergent series is a series that does not sum to a finite value. Understanding divergence is crucial in series analysis because it tells us the overall behavior of the series as it approaches infinity.

In the exercise, we've seen the series \( \sum_{n=1}^{\infty}\left(\frac{n+2}{n+1}\right)^{n} \) diverges. This was concluded using the Root Test, which determined that the limit \( L = e > 1 \), classifying the series as divergent.

• Divergent series can arise when terms do not approach zero fast enough or grow in magnitude.
• Such series lack a finite sum and often extend towards infinity.
• Recognizing divergent behavior helps in differentiating finite sums from those that continue indefinitely.

Applying these series tests provides deeper insights into the behavior of complex sequences and series.

Series Analysis

Series analysis involves a variety of methods to determine whether a series converges or diverges. Understanding the convergence or divergence of a series helps us understand the behavior and potential value a series can express.

In the given problem, multiple tools and methods, including the Limit Comparison Test and Root Test, were at our disposal. Each method serves to offer insights into different types of series and their behavior:

• Identifying key patterns in the terms of the series is crucial.
• Simplifying terms often involves algebraic manipulation, such as factoring and exponential approximation.
• The goal is to setup comparisons or compute limits that reveal how series behave as \( n \to \infty \).
• Not every test will provide an answer immediately — some may lead to new insights on how to proceed with the problem.

Effectively analyzing a series empowers students to deconstruct complex series into more manageable parts, ultimately leading to a better understanding of mathematical convergence.