Find  the sum to n terms of the series 1/1×3+1/3×5+15×7+.....

Let T be the rth term of the given series. Then, Tr=1(2r−1)(2r+1),r=1,2,3…n=1212r−1−12r+1 Hence, the required sum is ∑i=1n Tr=1211−13+13−15+15−17+…...+12n−1−12n+1=121−12n+1=n2n+1