text: how to prove that series (cos1w / 1 + cos2w / 2 + ... + cosnw / n), n=infinity, diverges or converges?   cos kw/k is real part of   (e^i(kw)/k)  because  e^ix = cos x + i sin x hence sum is real part of  sum ( e^i(kw)/k  the above is less than the term e^&#92;(ikw) which is geometric series  the series diverrges when e^i(kw) = 1 for all k else it converges  for the case e^ikw = 1  we get 1+1/2+1/3........... this diverges  for others it converges  when cos w = +/- 1 or w = npi divergent for other w convergent