For n = 1, 2, 3, the LHS has successive values of:

(1 – 1/2)

(1 – 1/2) + (1/3 – 1/4)

(1 – 1/2) + (1/3 – 1/4) + (1/5 – 1/6).

At each stage, two terms are added.

while the RHS has successive values of:

1/2

1/3 + 1/4

1/4 + 1/5 + 1/6.

At each stage, two terms are added at the right hand end, and one is removed from the left hand end.

When n is increased to n + 1, the LHS changes by:

1/[2(n + 1) – 1] – 1/2(n + 1)

= 1/(2n + 1) – 1/2(n + 1)

= [2(n + 1) – (2n + 1)] / 2(n + 1)(2n + 1)

= 1 / 2(n + 1)(2n + 1).

Under the same conditions, the RHS changes by:

1/(2n + 1) + 1/2(n + 1) – 1/(n + 1)

= [2(n + 1) + (2n + 1) – 2(2n + 1)] / 2(n + 1)(2n + 1)

= (2n + 2 + 2n + 1 – 4n – 2) / 2(n + 1)(2n + 1)

= 1 / 2(n + 1)(2n + 1).

Thus the LHS and the RHS are still equal, and hence by induction, the formula is proved.