Ʃ cos(k*pi)/k from 1 to infinity.
This is a test for convergence.
and when is the proper time to use the alternating series test
like using it on (-1)k(4k/8k) would result to divergence
since lim of (4k/8k) is infinity and not 0 but the function is really
convergent by the geometric series?
The Attempt at a Solution
Is it right to do this:
for k is odd cos is negative and for k is even cos is positive
cos(k*pi)\k < (-1)k/k
and by alternating series test ;
(-1)k*(1/k) since 1/k is decreasing and lim as 1/k approaches infinity is 0 then
(-1)k1/k converges thus cos(k*pi)/k converges by direct comparison test.