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