Question

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

Short answer

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

Step-by-step solution

Identify the function

Identify the function \(f(n)\) from the series. Here, \(f(n) = \frac{2n}{n^{2} + 1}\).

Check the conditions of the integral test

Make sure that the function \(f(x) = \frac{2x}{x^{2} + 1}\) satisfies the conditions for the integral test. It is continuous, positive for \(n \geq 1\), and decreasing, therefore we can apply the Integral test.

Compute the integral

We evaluate the integral of \(f(x)\) over the interval \([1, \infty)\). Using the substitution method where \(u = x^{2} + 1\), \(du = 2xdx\), we get \(\int_{1}^{\infty} \frac{2x}{x^{2}+1} dx = \int_{2}^{\infty} \frac{1}{u} du\). The integral evaluates to \(ln|u|\) evaluated from 2 to infinity, which is \(ln|2 - 1| = ln(1) = 0\).

Conclude the series convergence

Since the integral is finite, we conclude by the Integral Test that the series \(\sum_{n=1}^{\infty} \frac{2n}{n^{2}+1}\) converges.