How do you do this induction proof?

Question

Σ(n , i=1) 1/(2i-1)(2i+1) = n/(2n+1)

thanks in advance!!

Answer 1

Assume it's true for n=k.  Then the sum from 1 to k+1 should be

[k/(2k+1)], plus the (k+1)st term in the series:

[k/(2k+1)] + [ 1 / (2(k+1)-1)(2(k+1)+1)]

Simplify this and you should get (k+1) / (2(k+1) + 1).  Then show that the original summation holds true for n=1.