Sums of series for Riemann integrals

ThomasR

Hello

I'm having some difficulty in finding sums which relate to Riemann integrals.

The first one seems pretty simple.. a finite calculation of what would otherwise be the harmonic series i.e. 1/k from k=n to k=(2n-1). I can't see an easy way of finding a formula in terms of n, however.

The second one is similar.. sum of k/[n(k+n)] from k=1 to k=n. Again, completely stumped here as to how to get it into a nice format.

Finally, I need to sum from k=1 to k=(n-1) the values of sin(k*pi/2n). I have a similar result for cos(k*pi/2n) and it mentions using trigonometric/geometric series (which I presume means using the fact that sin(k*pi/2n) = Im(e^i(k*pi/2n)) but am unsure where to go from here.

Help on any of the above would be appreciated .. or just some general approach.

Cheers
Tom