I am supposed to prove the following by mathematical induction:

1/1*3 + 1/ 3*5 + ... + 1/[ (2n-1) * (2n+1) ] = n / (2n + 1)

I know that I have to prove

- The Basis Step: That P(1) is in S
- The inductive Step: That P(n) and P(n+1) is in S.

I already proved the Basis Step, but I am not sure if I am setting up the Inductive step correctly. Here's my work:

[ 1/ 1*3 + 3/ 3*5 + ... 1/ (2k - 1 ) (2k +1) = k/ (2k + 1)From here, I just manipulate the equations to get some form of (k+1)'s, but

[ "Same as above" + ... 1/ (2k - 1 ) (2k + 1) + 1/ [ ( 2 (k+1) -1 ) ( 2 (k+1) +1) = k/(2k + 1) + 1/ [ ( 2 (k+1) -1 ) ( 2 (k+1) +1)

did I set this up right?