Question

find the formula for the sum of 1/1*3+1/3*5+...........+1/(2n+1)(2n-1)​ = ?

Answer 1

Answer:

Since each term in this series is a product of three consecutive odd numbers, we may say,

Tn=1(2n−1)(2n+1)(2n+3)

Now, let's decorate it a little,

Tn=142n+3−2n+1(2n−1)(2n+1)(2n+3)

Tn=14[1(2n−1)(2n+1)−1(2n+1)(2n+3)]

Now, let,

Un=1(2n−1)(2n+1)

Which means,

Un+1=1(2n+1)(2n+3)

Now, we may say,

Tn=14[Un−Un+1]

∑nk=1Tk=14[∑nk=1Uk−Uk+1]

∑nk=1Tk=14[U1−Un+1]

∑nk=1Tk=14[11⋅3−1(2n+1)(2n+3)]

Which is the required sum.