/sci/ - Science & Math

Hey /sci/. I had a homework question the other day that was asking me to find whether or not a series converged. The series was:

<span class="math">\sum_{n=1}^\infty \frac{1}{n^2+1}[/spoiler]

And I could only use a divergence test (which would clearly fail), a p-series test (which couldn't be done in that form) or an integration test. I went with the integration test.

So, I set this up as:

<span class="math"> \int_1^\infty \frac{1}{n^2+1}[/spoiler]

Now, I didn't have a formula sheet in front of me, so I found, using Wolfram Alpha, that the indefinite integral integrates to arctan(x). Using wolfram alpha with the definite integral I just gave yields pi/4, which is fine and all and shows that the series converges, but how would I have been able to find that out without Wolfram Alpha?

Basically, I would need a way to figure out how to find the upper limit on arctan(x) as it approaches infinity. Is it something I just have to memorize? Cause I'd much rather rely on algebra than memorization. And of course I could have typed arctan(1) into my calculator to find that it equals pi/4, but we're not allowed to use calculators on tests. Hopefully my teacher wouldn't fuck us by putting a question like this on a test, but still, I'd rather be prepared.

Or is it possible to do this another way? I'd probably rather use a limit comparison test on this, but given the three options, and knowing the divergence test would fail, could I somehow get this to fit a p-series form?

>>3253729
good one

i don't understand the notion used in (d)...and someone explain to me what tg is?

solved in less than 22 min or it's not solvable.

I think not OP.

Let's get some jokes going here


pic unrelated

pots itt he who agrees