Question

Test the following series for convergence or divergence. Decide for yourself which test is easiest to use, but don't forget the preliminary test. Use the facts stated above when they apply. $$\sum_{n=1}^{\infty} \frac{n^{2}-1}{n^{2}+1}$$

Short answer

The series diverges by the Preliminary Test because the limit of the terms is 1, not 0.

Step-by-step solution

Preliminary Test

First, apply the preliminary test by evaluating the limit of the term as \( n \) approaches infinity. Compute \( \lim_{{n \to \infty}} \frac{{n^2 - 1}}{{n^2 + 1}} \).

Simplify the Limit

Simplify the expression under the limit: \( \lim_{{n \to \infty}} \frac{{n^2 - 1}}{{n^2 + 1}} = \lim_{{n \to \infty}} \frac{n^2(1 - \frac{1}{n^2})}{n^2(1 + \frac{1}{n^2})} = \lim_{{n \to \infty}} \frac{1 - \frac{1}{n^2}}{1 + \frac{1}{n^2}} \).

Evaluate the Limit

As \( n \) approaches infinity, both \( \frac{1}{n^2} \) terms approach 0. Therefore, \( \lim_{{n \to \infty}} \frac{1 - \frac{1}{n^2}}{1 + \frac{1}{n^2}} = \frac{1 - 0}{1 + 0} = 1 \).

Conclusion of Preliminary Test

Since the limit of the terms \( \frac{{n^2 - 1}}{{n^2 + 1}} \) as \( n \) approaches infinity is 1 (not equal to 0), the series \( \sum_{{n=1}}^{{\infty}} \frac{{n^2 - 1}}{{n^2 + 1}} \) diverges by the Preliminary Test.

Key concepts

Preliminary Test

Before diving into more intricate tests, we start with the preliminary test. This test checks the behavior of the terms in the series as they approach infinity. For the series to potentially converge, the terms must approach zero. If they do not, the series definitely diverges.

For the series \(\bigg\sum_{{n=1}}^{{\text{∞}}} \frac{{n^2 - 1}}{{n^2 + 1}}\), we analyze the limit of \(\frac{{n^2 - 1}}{{n^2 + 1}}\) as \(n \rightarrow \text{∞}\) . We simplify this to:
\(\frac{n^2(1 - \frac{1}{n^2})}{n^2(1 + \frac{1}{n^2})} \).
This further reduces to \(\frac{1 - \frac{1}{n^2}}{1 + \frac{1}{n^2}}\). As \(n\) becomes very large, \( \frac{1}{n^2} \) approaches zero, simplifying our limit to \( \frac{1}{1} = 1 \) .

In this case, since the limit is 1 (not equal to 0), the preliminary test tells us the series \(\bigg\sum_{{n=1}}^{{\text{∞}}} \frac{{n^2 - 1}}{{n^2 + 1}}\) diverges.
Hence, we can conclude that the series will not converge.

Limit Evaluation

Evaluating the limit is a core part of testing the convergence of a series. For our preliminary test, determining the behavior of the terms as \(n\) grows larger is crucial.

For the expression \(\frac{{n^2-1}}{{n^2+1}}\), we start by dividing the numerator and the denominator by \(n^2\). This allows us to simplify the term and make it easier to evaluate the limit.

The simplified form \( \frac{1-\frac{1}{n^2}}{1+\frac{1}{n^2}} \) shows us that as \(n \rightarrow \text{∞}\), both \( \frac{1}{n^2} \) in the numerator and denominator approach zero. Therefore, the entire fraction approaches \( \frac{1}{1} = 1\).

Since this limit is not zero, it indicates that the terms of the series do not diminish to zero, a prerequisite for the series to converge. Therefore, the series diverges. Understanding limit evaluation helps in recognizing how terms behave, simplifying complex fractions, and ultimately deciding convergence or divergence.

Infinite Series

An infinite series is the sum of the terms of an infinite sequence. Understanding whether such a series converges or diverges is fundamental in calculus.

\( \bigg\sum_{{n=1}}^{{\text{∞}}} \frac{{n^2 - 1}}{{n^2 + 1}} \) represents an infinite series where we sum terms indefinitely. For a series to converge, the sum must approach a finite value as we keep adding more terms.

If the sum goes to infinity or fails to settle on a definite amount, the series diverges. One of the first steps is to check individual term behavior, using tests such as the preliminary test and limit evaluation.

In our example, the series diverges because the terms do not diminish to zero. Understanding these concepts, and how to apply related tests, will help you navigate through more complex infinite series evaluations in your studies.