Question

Use the Integral Test to determine the convergence or divergence of the series. $$ \sum_{n=1}^{\infty} \frac{n}{n^{4}+1} $$

Short answer

The series \(\sum_{n=1}^{\infty} \frac{n}{n^{4}+1}\) converges.

Step-by-step solution

Define the function

First, define the continuous function \(f(x) = \frac{x}{x^{4}+1}\), which corresponds to the terms of the given series.

Show the function is positive and decreasing

Verify that \(f(x)\) is positive and decreasing. To prove that it is decreasing, compute its derivative \(f'(x)\). If \(f'(x) < 0\) for all \(x \geq 1\), then it proves that the function is indeed decreasing. So, \(f'(x) = \frac{1 - 4x^{4}}{(x^{4} + 1)^{2}}\). Since the numerator is less than 0 for \(x \geq 1\), the function is decreasing.

Apply the Integral Test

Now, apply the Integral Test by computing the integral of \(f(x)\) from 1 to infinity. \(\int_{1}^{\infty} \frac{x}{x^{4}+1} dx\). The result of this integral will determine if the series converges or diverges.

Evaluate the integral

Compute the integral \(\int_{1}^{\infty} \frac{x}{x^{4}+1} dx\). The integral equals to \(\frac{1}{4}\ln|\frac{x}{x^{4}+1}|\rvert_1^{+\infty}\). This integral equals to 0, (since as x approaches infinity the result is 0 and substituting x=1 gives us 0 as well), hence is finite.

Conclude the convergence or divergence

The integral test states that the series converges if the integral is finite. As computed in step 4, the integral \(\int_{1}^{\infty} \frac{x}{x^{4}+1} dx\) is finite, so the series \(\sum_{n=1}^{\infty} \frac{n}{n^{4}+1}\) converges.