# How do you use the Nth term test on the infinite series sum_(n=2)^oon/ln(n) ?

Aug 24, 2014

The Nth Term Test is a basic test that can help us figure out if an infinite series is divergent. It states that if the ${\lim}_{n \to \infty}$ of our series is not equal to $0$, the series is divergent. Note that this does not mean that if the ${\lim}_{n \to \infty} = 0$, the series is convergent, only that it might converge. All we can tell from this test is whether or not it diverges.

Using this test with our series, we have:

${\lim}_{n \to \infty} \frac{n}{\ln} \left(n\right)$

If we replace $n$ with $\infty$, we end up with:

$\frac{\infty}{\ln} \left(\infty\right) = \frac{\infty}{\infty}$

Since we have an Indeterminate Form of $\frac{\infty}{\infty}$, we can apply L'Hôpital's rule, which says that if we end up with $\frac{\infty}{\infty}$ or $\frac{0}{0}$, we can then take the $\lim$ of the derivative of the numerator over the derivative of the denominator. Since the derivative of $n$ is $1$, and the derivative of $\ln \left(n\right)$ is $\frac{1}{n}$, we have:

${\lim}_{n \to \infty} \frac{1}{\frac{1}{n}} = {\lim}_{n \to \infty} n = \infty$

Because we end up with a non-zero answer, we know that the series diverges.