Question

Use the limit comparison test to determine whether each of the following series converges or diverges. $$ \sum_{n=1}^{\infty} \frac{1}{n}\left(\tan ^{-1} n-\frac{\pi}{2}\right) $$

Short answer

The series converges by the limit comparison test with \( \sum_{n=1}^{\infty} \frac{1}{n^2} \).

Step-by-step solution

Identify the series for comparison

We need to choose a simpler series to compare with our given series. Usually, for series like given, we compare with a common p-series or geometric series. Here, a good choice is the harmonic series \( \sum_{n=1}^{\infty} \frac{1}{n} \), which diverges.

Simplify the given series

The series given is \( \sum_{n=1}^{\infty} \frac{1}{n}\left( \tan^{-1} n - \frac{\pi}{2} \right) \). As \( n \to \infty \), \( \tan^{-1} n \to \frac{\pi}{2} \), so \( \tan^{-1} n - \frac{\pi}{2} \to 0 \). Hence, this term \( \tan^{-1} n - \frac{\pi}{2} \sim -\frac{1}{n} \) for large \( n \).

Establish limit for comparison test

The limit comparison test suggests that if \( a_n \sim c b_n \) for some positive constant \( c \), the behavior of the series will be similar. Let \( a_n = \frac{1}{n}(\tan^{-1} n - \frac{\pi}{2}) \) and \( b_n = \frac{1}{n^2} \) (from our simplification in Step 2). Calculate \( \lim_{n \to \infty} \frac{a_n}{b_n} \).

Calculate the limit

We have \[ \lim_{n \to \infty} \frac{a_n}{b_n} = \lim_{n \to \infty} \frac{\frac{1}{n}(\tan^{-1} n - \frac{\pi}{2})}{\frac{1}{n^2}} = \lim_{n \to \infty} n(\tan^{-1} n - \frac{\pi}{2}) \approx \lim_{n \to \infty} \left(-\frac{1}{1+n^2}\right) = 1.\]

Conclude with the limit comparison test result

The limit calculated in Step 4 is 1, a positive constant, a key requirement of the limit comparison test. Since the comparison series \( \sum_{n=1}^{\infty} \frac{1}{n^2} \) converges, by the limit comparison test, the original series \( \sum_{n=1}^{\infty} \frac{1}{n}(\tan^{-1} n - \frac{\pi}{2}) \) converges as well.

Key concepts

Series Convergence

Series convergence is an essential concept in mathematics, determining whether an infinite series sums to a finite value or not. In simple terms, when we add up the elements of a series, we want to know if the total "stays put" at some number, or if it keeps growing indefinitely. Knowing about series convergence is important in many fields like physics and engineering, where it helps in predicting behaviors in systems over time.

When evaluating convergence, one of the tools we use is the Limit Comparison Test, which allows us to compare a given series with a known benchmark series. If our series behaves similarly near infinity to a well-understood convergent or divergent series, we can infer the convergence behavior of our series. This approach can simplify the problem significantly. The test can effectively show whether a series reaches, surpasses, or never gets to a certain value threshold.

Harmonic Series

The harmonic series is a famous example in calculus of a divergent series. It's written as \( \sum_{n=1}^{\infty} \frac{1}{n} \), and despite its terms getting smaller continuously, the total sum does not converge to a finite number—it grows infinitely large.

Because its divergence is well-established, it is often used in comparison tests, like the Limit Comparison Test. By leveraging the known behavior of the harmonic series, we can determine the behavior of more complicated series. Even when a series looks complex, understanding its relationship to a simple series like the harmonic one can illuminate whether it converges or not.

Arctangent Function

The arctangent function, denoted \( \tan^{-1}(x) \), is known for its role in trigonometry, giving the angle whose tangent is \( x \). It has a limiting behavior as \( x \to \infty \), approaching \( \frac{\pi}{2} \). This feature makes it useful in series to understand asymptotic behaviors—how a function behaves as values get very large or very small.

In the context of series convergence, the behavior \( \tan^{-1}(n) - \frac{\pi}{2} \to 0 \) as \( n \to \infty \) informs the convergence characteristics of an associated series. As demonstrated in the solution steps, determining that this expression becomes negligible for large \( n \) helps to transform the series into something more recognizable and simpler to analyze.

Mathematical Limits

Mathematical limits are a fundamental tool in calculus, used to describe the behavior of functions as they approach certain points or infinity. We often use limits in analysis to bridge the gap between finite mathematical constructs and "infinite" behaviors—like analyzing series convergence.

In the exercise problem, we calculated a limit to apply the Limit Comparison Test. By setting up our terms and calculating the limit of their quotient as \( n \to \infty \), we determine whether the original series behaves like our comparison series. Understanding limits helps practitioners to draw conclusions about elusive infinite processes by grounding them in more tangible, finite contexts.