Question

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

Answer 1

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->oo)` of our series is not equal to `0`, the series is divergent. Note that this does not mean that if the `lim_(n->oo) = 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->oo) n/ln(n)`

If we replace `n` with `oo`, we end up with:

`oo/ln(oo) = oo/oo`

Since we have an Indeterminate Form of `oo/oo`, we can apply L'Hôpital's rule, which says that if we end up with `oo/oo` or `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(n)` is `1/n`, we have:

`lim_(n->oo) 1/(1/n) = lim_(n->oo) n = oo`

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