Does this sequence tend to infinity?

a_n = n^4 |cos(n)|^4

Obviously it's unbounded but how can we tell if it goes to infinity or not? Am I right in thinking |cos(n)| can be infinitely small but never 0, since a natural number is never going to be a multiple of pi/2? And of course n^4 gets infinitely large so how can we decide if it goes to infinity, is there some clever trick? This is really bugging me!

Re: Does this sequence tend to infinity?

Quote:

Originally Posted by

**Magus01** a_n = n^4 |cos(n)|^4

Obviously it's unbounded but how can we tell if it goes to infinity or not? Am I right in thinking |cos(n)| can be infinitely small but never 0, since a natural number is never going to be a multiple of pi/2? And of course n^4 gets infinitely large so how can we decide if it goes to infinity, is there some clever trick? This is really bugging me!

If you can find arbitrarily large values of n such that $\displaystyle |n\cos n|<1$ then clearly the sequence will not tend to infinity. So we need to look at when $\displaystyle |\cos n|<\tfrac1n.$

To get $\displaystyle \cos n$ sufficiently small, we need n to be within approximately 1/n of an odd multiple of $\displaystyle \pi/2.$ Thus the question becomes: can we find arbitrarily large values of n for there exists an integer k such that $\displaystyle \bigl|n-\bigl(k+\tfrac12\bigr)\pi\bigr|<\tfrac1n,$ or equivalently $\displaystyle \Bigl|\frac n{\bigl(k+\tfrac12\bigr)} - \pi\Bigr|<\frac1{n\bigl(k+\tfrac12\bigr)}$ ? That is a question about rational approximations of $\displaystyle \pi$, and problems like that are not easy. You can find a bit more about this in my comments #5 and #10 in this thread.

Re: Does this sequence tend to infinity?

Interesting, thanks for the reply. Out of interest, do you actually know the answer?

It's funny because I was writing random sequences on the board to see how well my tutees understood properties of sequences, and I realised I actually couldn't answer this one. My initial instinct was that it did tend to infinity, since it seems like we should be able to take n large enough to "counteract" the small values |cos(n)| can take. But I can see that this could easily be wrong if |cos(n)| can periodically take arbitrarily small values.