Homework Help: Simple series question.

1. Dec 8, 2007

frasifrasi

Question asks to evaluate sum from 1 to infinity of n*cos(1/n) over (2n+1)

--> I am not sure how to do this.

I tried simplifying it to n/(2n+1) since the cos term is approaching 1 as n --> infinity. If i take the limit for that i get 1/2, so I concluded the series diverges.
--> but the bottom term is bigger than the top term, so why wouldn't it converge?

Thank you(it has been a while since I learned this so I am a bit confused).

2. Dec 8, 2007

Dick

Why would it converge just because the bottom term is larger than the top? You've already reached the correct conclusion. The limit of the nth term is 1/2. The sum can't possibly converge.

3. Dec 8, 2007

frasifrasi

I am saying, if i look at n/(2n+1), as n gets higher, the bottom is growing faster, so wouldn't it go to 0?

4. Dec 8, 2007

Dick

There is nothing wrong with posing theories like "If the denominator of a fraction is growing faster than the numerator then the limit is zero." We are doing mathematics here, so the next step is to test it. Take some sample terms like n=1000. 1000/2001. n=1000000. 1000000/2000001. They don't look like they are going to zero to me. It looks like they are approaching 1/2. So your 'theory' must be wrong. Try it with n/(2n).