Question

CALCULUS 2:
Could someone please help me determine which of the following is true for this ( http://www.tiikoni.com/tis/view/?id=ce7e84b ) infinite series?


A. This series converges.

B. This series diverges.

C. The integral test can be used to determine convergence of this series.

D. The comparison test can be used to determine convergence of this series.

E. The limit comparison test can be used to determine convergence of this series.

F. The ratio test can be used to determine convergence of this series.

Answer 1

G. The alternating series test can be used to determine convergence of this series.

Answer 2

I think only A, D and E being true is the answer, but why isn't that the case?

Answer 3

Actually, are the only two correct statements A and D?

Answer 4

The max it can be is 5/n^2 at any point, so I believe you can use the integral test.

Answer 5

Here's why I think you CANNOT use the integral test: The function is not monotonically decreasing.

Answer 6

Which is a requirement.:
http://en.wikipedia.org/wiki/Integral_test#Statement_of_the_test

http://www.wolframalpha.com/input/?i=%7B(4+%E2%80%93+cos(x%5E2))%2Fx%5E2%7D

Answer 7

You're wrong, this is a monotonically decreasing function. It's probably not what you're used to seeing because there's a little bit of wobbling going on from the cosine term, but each successive term is less than the previous term and is positive.

Answer 8

http://en.wikipedia.org/wiki/File:Monotonicity_example2.png

http://en.wikipedia.org/wiki/Monotonic_function

Answer 9

Actually, I think I'm right because, looking at the Wolfram Alpha link I posted above, the more-zoomed-in plot shows that it goes down then up frequently (which means it is not monotonically decreasing).

Answer 10

To state what a a monotone function is, to my understanding, it is a function that at least stays at the same dependent value as the independent value grows or goes always up or always down.

Answer 11

(When I was talking about the always-up or always-down part, I was talking about the dependent variable.)

Answer 12

Let's think about this. Why am I saying it's monotonically decreasing? What you've entered is a continuous graph of this, but what's really going on is we're only looking at discrete points. The top part of the fraction will oscillate between 3 and 5 only. The top will never change more than that, so we can make an integral test where we just look at 5 being on top since 3, 4, 5, or whatever 3-cos(n^2) is will always be less than or equal to 5.

The bottom on the other hand is n^2. That will actually decrease the fraction as you get larger values of n, so the whole discrete sum will fall under the integral which converges.

Answer 13

But the problem with that is that you seem to be using the direct comparison test along with the integral test, instead of the integral test alone.

If I understood correctly, you're saying that we can choose for example 3 as the numerator and then we can just deal with the 3 over the n^2, but that's combining two tests.

Answer 14

Maybe so, It's been over a year since I've taken cal 2 and I can honestly say I don't remember the names of tests. My philosophy is that it doesn't matter what something's called, all that matters is if you can apply the techniques. Good luck, it sounds like you pretty much know this more precisely than me about this haha.

Answer 15

@Kainui, thanks for this discussion ... debating this is helpful. :)

Answer 16

Having said that, if someone else could confirm or invalidate if it's only A and D that are true statements, I would really appreciate it!