Question

Use the integral test to determine whether the following sums converge. \(\sum_{n=1}^{\infty} \frac{n}{1+n^{2}}\)

Short answer

The series diverges.

Step-by-step solution

Identify the Function

The sum given is \( \sum_{n=1}^{\infty} \frac{n}{1+n^{2}} \). We need to identify a function \( f(x) = \frac{x}{1+x^{2}} \) corresponding to the terms of the series for applying the integral test.

Verify Conditions of the Integral Test

For the integral test, the function \( f(x) \) must be continuous, positive, and decreasing on \([1, \infty)\). Let's check these conditions:- **Continuous**: \( f(x) = \frac{x}{1+x^{2}} \) is continuous for all \( x \geq 1 \). It is not undefined at any point in this interval.- **Positive**: For \( x \geq 1 \), \( \frac{x}{1+x^{2}} > 0 \) since both the numerator and denominator are positive.- **Decreasing**: To verify that \( f(x) \) is decreasing, calculate the derivative \( f'(x) \) and show it is non-positive for \( x \geq 1 \).

Find and Interpret the Derivative

Compute the derivative \( f'(x) \) to verify whether it is decreasing:\[ f'(x) = \frac{d}{dx}\left( \frac{x}{1+x^{2}} \right) \]Using the quotient rule:\[ f'(x) = \frac{(1)(1+x^{2}) - (x)(2x)}{(1+x^{2})^{2}} = \frac{1 + x^{2} - 2x^{2}}{(1+x^{2})^{2}} = \frac{1 - x^{2}}{(1+x^{2})^{2}} \]For \( x \geq 1 \), \( 1 - x^{2} \leq 0 \), indicating \( f'(x) \leq 0 \). Thus, \( f(x) \) is a decreasing function.

Apply the Integral Test

Since \( f(x) \) satisfies the conditions, we apply the integral test by evaluating:\[ \int_{1}^{\infty} \frac{x}{1+x^{2}} \, dx \]We can solve this using a substitution: let \( u = 1+x^{2} \), \( du = 2x \, dx \), then \( x \, dx = \frac{1}{2} du \). Therefore:\[ \int \frac{x}{1+x^{2}} \, dx = \int \frac{1}{2} \frac{du}{u} = \frac{1}{2} \int \frac{du}{u} = \frac{1}{2} \ln|u| + C \]Replacing \( u = 1+x^{2} \):\[ \frac{1}{2} \ln(1+x^{2}) \Bigg|_{1}^{\infty} \]

Evaluate the Improper Integral

Compute \( \int_{1}^{\infty} \frac{1}{2} \ln(1+x^{2}) \, dx \) as a limit:\[ \lim_{b \to \infty} \left( \frac{1}{2} \ln(1+b^{2}) - \frac{1}{2} \ln(2) \right) = \lim_{b \to \infty} \frac{1}{2} (\ln[(1+b^{2})] - \ln(2)) \]As \( b \to \infty \), \( \ln(1+b^{2}) \to \infty \). Thus, the integral diverges.

Conclusion from the Integral Test

Because the integral diverges, by the integral test, the series \( \sum_{n=1}^{\infty} \frac{n}{1+n^{2}} \) also diverges.

Key concepts

Convergence of Series

The concept of convergence of series is fundamental in understanding whether an infinite series will sum to a finite number. A series \( \sum_{n=1}^{\infty} a_n \) converges if the sequence of its partial sums \( \{S_N\} \) approaches a definite limit as \( N \to \infty \). To determine convergence, various tests are used, with the Integral Test being one of them.

The Integral Test is particularly useful for series whose terms can be related to a function that is continuous, positive, and decreasing. By comparing the series to an improper integral of the corresponding function, you can infer the behavior of the series:

• If the integral of the function converges, then the series converges.
• If the integral diverges, the series diverges.

It's important to remember that convergence does not always imply the total sum can be easily calculated, only that it is finite.

Improper Integrals

Improper integrals are integrals where either the interval of integration is infinite or the integrand approaches infinity within the limits of integration. These integrals differ from regular definite integrals and require special techniques to evaluate.

To evaluate an improper integral like \( \int_{1}^{\infty} \frac{x}{1+x^{2}} \ dx \), you approach it as a limit:

• Express the improper integral through a limit: \( \int_{1}^{\infty} f(x) \, dx = \lim_{b \to \infty} \int_{1}^{b} f(x) \, dx \).
• Calculate the definite integral for \( x \) from 1 to \( b \) and then evaluate the limit as \( b \) approaches infinity.

Thus, the integral's divergence or convergence will determine the series' behavior in the context of the Integral Test.

Differentiation

Differentiation is a process used to find the rate at which a function is changing at any given point, frequently needed to analyze the behavior of functions related to infinite series. For the Integral Test, verifying that a function is decreasing is key.

In this context, differentiating the function \( f(x) = \frac{x}{1+x^{2}} \) gives:

• The derivative \( f'(x) = \frac{1 - x^2}{(1+x^2)^{2}} \) describes the slope of the tangent line at any point \( x \).
• The condition \( f'(x) \leq 0 \) demonstrates that \( f(x) \) is non-increasing for \( x \geq 1 \).

By showing \( f(x) \) decreases, it's confirmed that the Integral Test can be applied, highlighting the series' convergence or divergence.

Function Behavior Analysis

Analyzing the behavior of functions is crucial for applying the Integral Test. This analysis focuses on ensuring the function associated with the series meets necessary conditions. Here, \( f(x) = \frac{x}{1+x^{2}} \) is confirmed to be:

• Continuous: No discontinuities exist for \( x \geq 1 \), making the function seamless across the interval.
• Positive: Since both numerator and denominator are positive for \( x \geq 1 \), \( f(x) \) remains positive.
• Decreasing: As shown by the derivative being non-positive, \( f(x) \) is non-increasing.

Proper analysis ensures \( f(x) \) satisfies all criteria for the Integral Test, guiding us in concluding the series' divergence.