Question

Determine the convergence of the given series using the Root Test. If the Root Test is inconclusive, state so and determine convergence with another test. $$\sum_{n=1}^{\infty}\left(\frac{n^{2}-n}{n^{2}+n}\right)^{n}$$

Short answer

The series converges by the Limit Comparison Test with \(b_n = \left(1-\frac{2}{n}\right)^n\).

Step-by-step solution

Apply the Root Test Formula

The Root Test states that for the series \(\sum a_n\), we calculate \(L = \lim_{n \to \infty} \sqrt[n]{|a_n|}\). If \(L < 1\), the series converges absolutely; if \(L > 1\), the series diverges; and if \(L = 1\), the test is inconclusive. For this series, \(a_n = \left(\frac{n^2-n}{n^2+n}\right)^n\), so we find \(L = \lim_{n \to \infty} \left|\frac{n^2-n}{n^2+n}\right|\).

Simplify the Limit Expression

First, simplify \(\frac{n^2-n}{n^2+n}\) by dividing the numerator and the denominator by \(n^2\). This yields \(\frac{1-\frac{1}{n}}{1+\frac{1}{n}}\). The expression simplifies as \(n\) approaches infinity: \(\lim_{n \to \infty} \frac{1-\frac{1}{n}}{1+\frac{1}{n}} = 1\). Thus, \(L = \lim_{n \to \infty} \left(\frac{n^2-n}{n^2+n}\right)^n = 1^n = 1\).

Root Test Conclusion

Since the Root Test yields \(L = 1\), this means the Root Test is inconclusive in determining convergence of the series. We need to apply another test to check for convergence.

Consider another test - The Ratio Test

The series is difficult to deal with using the Ratio Test due to the powers involved, so let's use a more straightforward approach using the Limit Comparison Test.

Apply the Limit Comparison Test

Consider the limit comparison of \(b_n = \left(1-\frac{2}{n}\right)^n\). Finding \(L = \lim_{n \to \infty} \frac{a_n}{b_n}\), results in \(L = \lim_{n \to \infty} \left(\frac{1}{1+\frac{2}{n}}\right)^n = e^{-2}\). Since this is a positive finite number and \(b_n\) converges because it behaves like \(e^{-2}\), \(a_n\) converges as well.

Key concepts

Series Convergence

Series convergence tells us whether a sum of terms from a sequence results in a finite number or not. When we talk about a series \( \sum_{n=1}^{\infty} a_n \), we explore whether the total becomes a definite value as more terms are added. This is critical for understanding the nature of infinite sequences and series, as it helps us to gauge their overall behavior.
To determine convergence, various tests can be implemented. These tests rely on mathematical conditions to guide our understanding:

• If a series converges, the values added up reach a specific sum.
• If it diverges, the sum grows without bound or fails to approach a definite value.

Tests like the Root Test or Ratio Test are common tools used to analyze series convergence. Each test has different criteria and uses, providing mathematicians several methods for exploring infinite series behavior.

Limit Comparison Test

The Limit Comparison Test is a handy tool when dealing with series convergence. It simplifies the process by letting us compare the series in question to another series with known convergence behavior. Here's how it works:
Given two series \( \sum a_n \) and \( \sum b_n \, \) we find the limit \( L = \lim_{n \to \infty} \frac{a_n}{b_n} \. \) To draw conclusions from the Limit Comparison Test:

• If \( L \) is a positive finite number, and \( b_n \) converges, then \( a_n \) also converges.
• Similarly, if \( b_n \) diverges, \( a_n \) diverges as well, under the same \( L \).

In essence, this test can make a difficult problem easier by letting us make comparisons rather than working directly with complex terms.

Ratio Test

The Ratio Test is a method of determining series convergence by analyzing the ratio of subsequent terms. For a series \( \sum a_n \, \) here’s what you do:
Compute the limit of the absolute value of the ratio of consecutive terms: \( L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|. \)

• If \( L < 1, \) the series converges absolutely.
• If \( L > 1, \) the series diverges.
• If \( L = 1, \) the test is inconclusive, and another test must be applied.

This method is particularly useful when the series terms \( a_n \) form a pattern that simplifies upon dividing one term by the next. However, if \( L = 1, \) as happened in the given exercise, the Ratio Test does not provide a conclusive result, and other methods need to be considered.

Power Series

Power series are special kinds of series where each term involves powers of the same variable. A power series has the form: \( \sum_{n=0}^{\infty} c_n(x-a)^n, \) where \( c_n \) are coefficients and \( a \) is the center of the series.
Power series are essential in representing functions, especially because:

• They can approximate complex functions using polynomials.
• Their convergence pattern depends on values of \( x \).

Understanding the radius of convergence is crucial with power series; it's the range of \( x \) for which the series converges. The convergence behavior can change for different values of \( x \, \) which often requires detailed exploration using various convergence tests. Power series become particularly valuable in calculus and various fields of science and engineering due to their expansive utility.