Determine whether the following are convergent or divergent. If it is convergent find its sum. Make sure to fully justify all of your work.

$$\sum_{n=4}^{\infty} \arctan(\ln(n))$$

I have literally no idea. I've look at my notes from this weeks lecture, none of them deal with trig series. I've looked at the assigned textbook chapters none of them deal with trig. I think I can integrate this using parts but that won't find its sum which I also need. I've tried googling it but they all use weird trig identities I've never seen before. Can anyone help me get started?

The only test I know of is

geometric series test, integral test, divergence test, and telescoping test.

EDIT: I just realized this diverges because $\lim_{n\to\infty} \arctan(\ln(n)) = \pi/2$.

Anyone know how to properly say $\lim_{n\to\infty} \arctan(\ln(n))$ is $\pi/2$ ? I think i use sqeeze theorem? I completely forgot how to do this.


Here's a very quick proof:

  1. $\ln$ is a monotonically increasing function.
  2. $\arctan$ is a monotonically increasing function.
  3. Therefore, $\arctan(\ln x)$ is a monotonically increasing function.
  4. Therefore, the sum in question is bound from below by $\arctan(\ln 4) \times \sum_{i = 4}^\infty 1 = \infty$ (because each term after the first is larger than that value).
  5. Thus the sequence diverges.

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