I have this problem:

4/3-8/15+12/35-16/63+...+((-1)^(1/2N-1)*2N)/(N^2-1)=(N+1+(-1)^(1/2N+1))/(N+1)

Works for every whole n that is an even number.

I checked if it works when n=2 and it does.

I said that when n=k it works.

While proving that it works when n=k+2 I got it down all the way to this but can't figure out how to proceed:

(K+1+(-1)^(1/2K+1))/(K+1)-((1)^(1/2K-1/2)*(2K+2))/(K^2+4K+4)=(K+3+(-1)^(1/2K+2))/(K+3)

I also included an image so that it will be easier to read.

Please help and if you can include explanations of every step that would be great.