Problem 11 - Essential Calculus

Question

Use the Integral Test to determine the convergence or divergence of the series. $$ \sum_{n=1}^{\infty} \frac{\arctan n}{n^{2}+1} $$

Step-by-Step Solution

Answer

The series \( \sum_{n=1}^{\infty} \frac{\arctan n}{n^{2}+1} \) is convergent.

Step 1: Verify the conditions of the Integral Test

Define the function \( f(x) = \frac{\arctan x}{x^2+1} \). The integral test applies when \( f(x) \) is positive, continuous, and decreasing when \( x >1 \). Given that \(\arctan x \) increases monotonically from \(0\) to \( \frac{\pi}{2} \) for \( x \geq 0 \) and \( x^2+1 > 1 \), \( f(x) \) is a positive and decreasing function when \( x > 1 \). By the nature of the \(\arctan x \) function, \( f(x) \) is continuous. Thus, \( f(x) \) meets the conditions for the Integral Test.

Step 2: Apply the Integral Test

By the Integral Test, the convergence of the series is equivalent to the convergence of the integral \( \int_{1}^{\infty} \frac{\arctan x}{x^{2}+1} dx \). We can solve this definite integral by using a standard substitution method. Let \( u = \arctan x \) thus \( du = \frac{1}{1 + x^2} dx \) and our integral simplifies to \( \int_{\frac{\pi}{4}}^{\frac{\pi}{2}} u du \). Evaluating this gives \( \frac{\pi^2}{16} - \frac{\pi^2}{32} = \frac{\pi^2}{32} \).

Step 3: Conclusion

Because the integral from step 2 is finite, we can conclude, by the Integral Test, that the original series \( \sum_{n=1}^{\infty} \frac{\arctan n}{n^{2}+1} \) is convergent.

Key Concepts

Convergence

When understanding the concept of convergence, especially in the context of series, we are essentially asking whether the sum of an infinite series reaches a particular finite value instead of approaching infinity. Convergence is a fundamental concept in calculus and analysis. In simpler terms, if you add up an infinite number of terms in a series but the total can be assigned a particular finite value, the series converges.

When applying the Integral Test for convergence, we use it to determine if an infinite series converges by comparing it to an integral. If the integral \( \int_{1}^{\infty} f(x) \, dx \) converges, meaning it sums to a finite value, then the series \( \sum_{n=1}^{\infty} a_n \) given by \( a_n = f(n) \) also converges. This test is particularly useful when functions involve transcendental parts like the arctan function in this exercise.

Series

A series in mathematics is what we call the sum of the terms of a sequence. In this exercise, we are working with an infinite series, meaning we are adding up infinitely many terms.

Mathematically, a series can be written as \( \sum_{n=1}^{\infty} a_n \), where \( a_n \) are the terms of the series. The series in question, \( \sum_{n=1}^{\infty} \frac{\arctan n}{n^2+1} \), involves the arctan function in its terms.

• This type of series is crucial in analysis and often studied using various tests to determine convergence, such as the Integral Test.
• Understanding the behavior of series is fundamental in calculus, as they appear in numerous mathematical problems and practical applications.

Arctan Function

The arctan function, also known as the inverse tangent, is a special function that finds the angle whose tangent is a given number. It is denoted as \( \arctan x \) and is defined on all real numbers.

• This function increases monotonically from 0 to \( \frac{\pi}{2} \) as \( x \) moves from 0 to infinity.
• In terms of derivatives, \( \frac{d}{dx} \arctan x = \frac{1}{1+x^2} \), which is useful in substitution during integration, as seen in our exercise.

The arctan function plays a crucial role in trigonometry and calculus, especially when dealing with integrals and series, as it helps simplify solutions by substituting variables that lead to solvable standard forms.

Integral Calculus

Integral calculus is a part of mathematics that focuses on integrals and their properties, operations, and applications. It is one of the principal methods of solving calculus problems and is used extensively in physics, engineering, and statistics.
The main idea is to find the accumulation of quantities, which could mean areas under curves, total growth, or accumulated change.

• The Integral Test applied in this exercise is a technique that uses integral calculus to determine whether a series converges or diverges.
• As shown in the solution, converting a series into an integral allows us to evaluate the behavior of an infinite series by assessing the behavior of an improper integral.
• We saw this practical use of substitution in our example, transforming the integration of \( \frac{\arctan x}{x^2 + 1} \) into a simpler form \( \int u \, du \).

Integral calculus tools like these are invaluable both for theoretical mathematics and real-world problem-solving.