Question dealing with absolute convergence, conditional convergence, vs divergence !

Which of the following statements is true about the series

∞
∑ (-1)^n cos(1/n)
n=0

a) the series is absolutely convergent
b) the series is conditionally convergent
c) the series is divergent

how do I go about showing the work for this?

use limit,

when n--->infinity
1/n becomes zero

cos(0)=1

then for sufficiently large n you get

-1,+1,-1,+1, -1,+1,....

which goes on forever, it never converges to one value so it is divergent

Note that this is an alternating series. An alternating series converges if:
(a) The non-alternating part forms a monotonically decreasing sequence
(b) The non-alternating part goes to zero as n --> infinity.

Since cos(1/n) fails to meet both of these requirements, the series diverges.

I hope this helps!

c) the series is divergent because as n->∞, the general term oscillates between 1 and -1

This series diverges by n-th term test!