Confusion with integration of sums

[Nick Jackson]

Hello guys, since I am new at sums and multivariable calculus I faced a problem when I stumbled upon this: \sum_{r=0}^{k} \binom{n}{4r+1} x^{n-4r-1} y^{4r+1} = \sum_{r=0}^{b} \binom{n}{4r+3} x^{n-4r-3} y^{4r+3} Well, the problem is that I don't know if it's possible to put a limit in every part of the equation and then convert it to an integral (I am trying to prove that \forall x , y \in \mathbb{R}, \quad \exists n \in \mathbb{N} , n \in \mathbb{N} , n \neq 0 so that the relation holds). Can I do it or does it violate any rule? And if it is possible to do it how would the multivariable integration be? (If you want the relations between n, k, b just ask)

 

[Stephen Tashi]

The usual way to approach proving such a result is by mathematical induction. Two functions of N can approach the same limit as N approaches infinity without always being equal to each other. So proving the left and right hand sides have the same limit would not prove the two sides are equal for all values of N.

 

[Nick Jackson]

Oh i get it! Thanks a lot!