Re: Why is this limit 0?

[The World Wide Wade]

In article <quaTg.15$Zj3.12@trnddc03>, "TCL" <tlim1@xxxxxxxxxxx> 
wrote:

> Let  p(n)= int_0^pi  | sin(n+ 0.5)x| /  sin (0.5 x)  dx.
> I am trying to prove that
> lim    p(n)/ ln n   = 0

sin(x/2)/x is bounded on the interval, so we can instead consider 
int_0^pi  |sin((n+1/2)x)| /x dx = int_0^(n+1/2)pi  |sin(y)| /y 
dy. Now int_kPi^(k+1)Pi  |sin(y)| /y dy > 2/((k+1)Pi), so the 
integral has to grow at least as fast as sum(k=1,n-1) 2/((k+1)Pi) 
>= C*log(n).