How to prove that series (cos1w / 1 + cos2w / 2 + ... + cosNw / N), N=infinity, diverges or converges?

Answer 1

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^\(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

Answer 2

using the ratio test

let u(n)=cos(n)w/n
let u(n+1)=cos(n+1)w/(n+1)

u(n+1)/un
=(cos(n+1)w/cosnw)*n/(n+1)

hence,since n tends to infinity,n/(n+1)
tends to1,not 0 and (cos(n+1)w/cosnw)
alternates,therefore,this series is
divergent and is infinite

{it is rather like 1/2,1/3,1/4,1/5+........}

i hope that this helps

Answer 3

diverges an is infinity