Question

is the following alternating series divergent or convergent series?
(please show proof)

∞
∑ (-1)^(n-1) cos(1/n)
n=1

Answer 1

Set your series = a(n).
Then use a(n+1) like this:
|a(n+1)/a(n)|
Working this you will find eventually find:
cos(1/n+1)/cos(1/n)
which should evaluate to <1 and therefore be absolute convergent. I'm not used to evaluating series with trigonometric components though.

Answer 2

doesnt that go to 1 and not < 1 since we take the limit to infinity using the ratio test

Answer 3

You might be correct in that, I'm not a 100% sure, been a while since I did the infinite series. If it does evaluate to 1, you'd need to consider the different cases and evaluate from there.