Help with summation

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October 13th 2008, 10:06 AM
Hweengee

Help with summation

Given that
$3\,\sum _{m=1}^{M} \left( \cot \left( {\frac {m\pi }{2\,M+1}} \right) <br /> \right) ^{2}$

is a polynomial of degree r in M, find r and the coefficient of $M^r$ .(The coefficient is an integer)

Any ideas how to go about solving this problem?
October 13th 2008, 10:37 PM
Moo

Quote:

Originally Posted by Hweengee

Given that
$3\,\sum _{m=1}^{M} \left( \cot \left( {\frac {m\pi }{2\,M+1}} \right) <br /> \right) ^{2}$

is a polynomial of degree r in M, find r and the coefficient of $M^r$ .(The coefficient is an integer)

Any ideas how to go about solving this problem?

Using power series ? (Worried)
October 14th 2008, 03:16 AM
Hweengee

I have no idea. I am allowed to use MAPLE, but the command sum(expression,m=1..M) doesn't give a general solution in a polynomial form, so I am quite confused. And we haven't gotten to sequences and series in my calculus course yet.
October 14th 2008, 09:19 AM
Opalg

I have no idea how to prove it, but this formula is what you are looking for. It says $\sum_{k=1}^{n-1}\cot^2(k\pi/n) = \tfrac13(n-1)(n-2)$ . Put n=2M+1 and you get $3\sum_{k=1}^{2M}\cot^2(k\pi/(2M+1)) = 2M(2M-1)$ .

Notice that the sum goes from 1 to 2M. But using the fact that $\cot(\pi-x)=-\cot x$ you can see that this is twice the sum from 1 to M.

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