Let and and
I need the general formula for the following summation in terms of and :
Start by writing out some values and looking for a pattern:
It should be clear that if n is odd, then .
It should also be clear that if n is a multiple of 4, we have equal to divided by a product of even numbers from n down to 2: if n= 4i, then and we can treat that product as a factorial by factoring out "2":
If n is even but not a multiple of 4, n= 4i+ 2, then it is almost exactly the same except that the numerator is , of course, and there is no "2" in the product in the denominator. We can fix that by taking a "2" out of the denominator: by multplying by 2.
The whole sum can then be written as two separate sums:
For future reference, a product of odd numbers can be written by multiplying and dividing by the even numbers:
and then treating the product of even numbers in the denominator as above: