Problem 40

Question

In Exercises $37-40,$ find the series' radius of convergence. $$ \begin{array}{l}{\sum_{n=1}^{\infty}\left(\frac{n}{n+1}\right)^{n^{2}} x^{n}} \\\ {\text { (Hint: Apply the Root Test.) }}\end{array} $$

Step-by-Step Solution

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Answer

The series' radius of convergence is $ e $.

1Step 1: Understand the Root Test

The root test helps determine convergence of the series by considering the limit of the nth root of the absolute value of the general term. If $ \limsup_{n \to \infty} \sqrt[n]{|a_n|} < 1 $, the series converges absolutely.

2Step 2: Analyze the General Term

Here the general term of the series is $ a_n = \left( \frac{n}{n+1} \right)^{n^2} x^n $. We must take the nth root and eventually find the limit as $ n \to \infty $.

3Step 3: Apply the nth Root

Find $ \sqrt[n]{|a_n|} = \sqrt[n]{\left( \frac{n}{n+1} \right)^{n^2} x^n} = \left( \frac{n}{n+1} \right)^{n} |x| $.

4Step 4: Evaluate the Limit

Evaluate $ \lim_{n \to \infty} \left( \frac{n}{n+1} \right)^{n} $. Using the fact that $ \left(1 - \frac{1}{n+1} \right)^n \to e^{-1} $, the expression approaches $ \frac{1}{e} $.

5Step 5: Establish the Condition for Convergence

The series converges absolutely if $ \lim_{n \to \infty} \left( \frac{n}{n+1} \right)^{n} |x| < 1 $. Simplifying, this means $ \frac{|x|}{e} < 1 $.

6Step 6: Solve for the Radius of Convergence

For convergence, we need $ |x| < e $. Therefore, the radius of convergence, $ R $, is $ e $.

Key Concepts

Root TestSeries ConvergenceLimit Evaluation

Root Test

The Root Test is an essential tool for determining the convergence of an infinite series. It focuses on the behavior of the terms in the series as they approach infinity. To apply the Root Test, follow these simple steps:

Identify the general term of the series, denoted as $a_n$.
Calculate the $n$-th root of the absolute value of $a_n$, which is expressed as $\sqrt[n]{|a_n|}$.
Find the limit as $n$ approaches infinity: $\lim_{n \to \infty} \sqrt[n]{|a_n|}$.

Once the limit is known, apply these criteria:- If the limit is less than 1, the series converges absolutely.- If the limit is greater than 1, the series diverges.- If the limit is exactly 1, the test is inconclusive, and other methods may be needed to determine convergence.In our example with the series $\sum_{n=1}^{\infty}\left(\frac{n}{n+1}\right)^{n^2} x^n$, we discovered that the expression simplifies to $(\frac{n}{n+1})^{n} |x|$. By evaluating $\lim_{n \to \infty} (\frac{n}{n+1})^{n}=\frac{1}{e}$, we verified that the Root Test applies effectively here.

Series Convergence

Understanding the convergence of a series is crucial in many areas of mathematics. Convergence refers to whether the sum of an infinite series approaches a finite value. Different tests for convergence satisfy different series, and choosing the correct one often depends on the form of the series itself.

The key idea is to determine if the terms of the series become small enough so that adding them yields a finite result.A series specifically converges when:

The terms decrease in magnitude as you progress through the series.
The entire series sum approaches a particular number, often denoted as "the limit."

In our exercise, we used the Root Test to determine where the series converges absolutely, which means both the positive and negative values are being considered. For instance, the convergence condition $\frac{|x|}{e}<1$ dictates that \(|x|

Limit Evaluation

Evaluating limits is a fundamental concept in calculus, and it also plays a significant role in series convergence tests. When applying the Root Test, evaluating the limit of $ (\frac{n}{n+1})^n $ as $ n \to \infty $ is a required step. Getting this limit correct is crucial to accurately applying the convergence criteria.To solve limits involving expressions that tend to infinity, we often use known limit properties and approximations. In our case, a key property used is that\[\left(1 - \frac{1}{n+1} \right)^n \to e^{-1}\]as $ n $ becomes very large. This transformation simplifies complex expressions, making the limit evaluation possible. Here the limit evaluated at $ (\frac{n}{n+1})^n $ as $ n \to \infty $ results in $ \frac{1}{e} $, showing how this trick simplifies the step in the process.

Understanding and mastering these evaluations help solve series convergence problems with greater ease.

Problem 40

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