What is the simplest form of the sum 1/(1+square root 2) + 1/(square root 2+square root 3)+...........+1/(square root 2000+square root2001)?

We'll write the general term of the
sum:

1/(sqrt k +
sqrt(k+1))

We'll multiply the numerator and denominator by
the conjugate of denominator:

(sqrt k - sqrt(k+1))/[(sqrt
k)^2 - (sqrt(k+1))^2] = (sqrt k - sqrt(k+1))/(k - k -
1)

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

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

We'll put k=1 => 1/(1+sqrt2) = sqrt 2 -
1

We'll put k = 2=> 1/(sqrt 2+sqrt 3) = sqrt 3 -
sqrt
2

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

We'll
put k = 2000=> 1/(sqrt 2000+sqrt 2001) = sqrt 2001 - sqrt
2000

We'll add the
terms:

1/(1+sqrt2) + ... + 1/(sqrt 2000+sqrt 2001) = sqrt 2
- 1 + sqrt 3 - sqrt 2 + ... + sqrt 2001 - sqrt 2000

We'll
eliminate like terms:

1/(1+sqrt2) + ... + 1/(sqrt 2000+sqrt
2001) = sqrt 2001 - 1

The simplest form of
the given sum is: S = sqrt 2001 - 1.