Question

Use the Integral Test to determine the convergence or divergence of the series. $$ \frac{1}{3}+\frac{1}{5}+\frac{1}{7}+\frac{1}{9}+\frac{1}{11}+\cdots $$

Short answer

The series is divergent.

Step-by-step solution

Formulate the Function

Formulate the function \(f(x) = \frac{1}{2x+1}\) that matches the terms of the series when \(x\) is a non-negative integer. We start with \(f(x)\) because this function will produce the terms in our series when we substitute positive integers for \(x\).

Determine Convergence

This function should satisfy the three conditions that make it suitable for the Integral Test. It is continuous, positive, and decreasing on the interval \([0, \infty)\). So, we can proceed with the Integral Test.

Compute the Integral

Calculate the integral of the function from 0 to infinity. The integral is \(\int_0^\infty \frac{1}{2x+1} dx\). This is an improper integral that can be solved using the basic antiderivative of \(1/x\) which is \(ln|x|\). The result of the integral is \(\lim_ {t \to \infty} ln(2t+1) - ln(1)\). When evaluating this limit at infinity, it is found to be infinity.

Interpret the Result

The result of our integral from Step 3 is infinity. According to the Integral Test, if the integral of \(f\) from \(1\) to \(\infty\) is infinite, this means that original series is divergent.