The denominator of each fraction is the product of 2 consecutive numbers.

1/1*2 + 1/2*3 + ....+ 1/k(k+1)

We'll write this type of ratio 1/n(n+1) as the result of addition or subtraction of 2 elementary fractions:

We'll write the ratio:

1/n(n+1) = A/n + B/(n+1) (1)

We'll multiply the ratio A/n by (n+1) and we'll multiply the ratio B/(n+1) by n.

1/n(n+1) = [A(n+1) + Bn]/n(n+1)

We'll write the numerators, only.

1 = A(n+1) + Bn

We'll remove the brackets:

1 = An + A + Bn

We'll factorize by n to the right side:

1 = n(A+B) + A

We'll put equal the correspondent coefficientsfrom both sides.

A+B = 0

A = 1

1 + B = 0

B = -1

We'll substitute A and B into the expression (1):

**1/n(n+1) = 1/n - 1/(n+1) (2)**

According to (2), we'll give values to n:

for n = 1 => 1/1*2 = 1/1 - 1/2

for n = 2 => 1/2*3 = 1/2 - 1/3

.....................................................

for n = k-1 => 1/k(k-1) = 1/(k-1) - 1/k

for n = k => 1/k(k+1) = 1/k - 1/(k+1)

If we'll add the ratios from the left side, we'll get:

1/1*2 + 1/2*3 + ... + 1/k(k-1) + 1/k(k+1) = 1/1 - 1/2 + 1/2 - 1/3 + ..... + 1/(k-1) - 1/k + 1/k - 1/(k+1)

We'll eliminate like terms from the right side:

1/1*2 + 1/2*3 + ... + 1/k(k-1) + 1/k(k+1) = 1 - 1/(k+1)

S = (k + 1 - 1)/(k+1)

We'll eliminate like terms and we'll get the final result of the sum:

S = k /(k+1)