Given the string xn=Sum 1/k(k+1)(k+2) where k = 1 to n , what is xn ?

To determie the general terms of the string xn, we'll have
to decompose the given fraction into elementary ratios
first.

1/k(k+1)(k+2) = A/k + B/(k + 1) + C/(k +
2)

We'll multiply both sides by
k(k+1)(k+2):

1 = A(k+1)(k+2) + Bk(k+2) +
Ck(k+1)

We'll remove the
brackets:

1 = Ak^2 + 3Ak + 2A + Bk^2 + 2Bk + Ck^2 +
Ck

We'll combine and factorize like terms from the right
side:

1 = k^2(A + B + C) + k(3A + 2B + C) +
2A

Comparing, we'll get:

2A
=1

A =
1/2

A+B+C = 0

B
+ C = -1/2 (1)

3A + 2B + C =
0

2B + C = -3/2 (2)

We'll
subtract (1) from (2):

2B + C - B - C = -3/2 +
1/2

We'll combine and eliminate like
terms:

B =
-1

C = -1/2 -
B

C = 1 - 1/2

C
= 1/2

1/k(k+1)(k+2) = 1/2k - 1/(k + 1) +
1/2(k + 2)

Now, we'll determine the
sum:

For k = 1:

1/2 - 1/2 +
1/6

For k = 2:

1/4 - 1/3 +
1/8

For k = 3:

1/6 - 1/4 +
1/10

For k = 4:

1/8 - 1/5 +
1/12

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

For
k = n:

1/2n - 1/(n + 1) + 1/2(n +
2)

We'll add all terms and we'll
get:

(1/2)(1 + 1/2 + 1/3 + ... + 1/n) - 1 - 1/2 - (1/3 +
... + 1/n + 1/(n + 1)) -  + (1/2)(1/3 + ... + 1/n + 1/(n +
1))

Combining the terms, we'll
get:

xn = (1/2)[1/2 - 1/(n + 1)(n +
2)]