Question

Determine the value of the sum​ along with the steps

Answer 1

The n-th term in the sum (where n is a natural number 1 ≤ n ≤ 1999) is

`\sqrt{1 + \frac1{n^2} + \frac1{(n+1)^2}}`

and can be simplified as

`\sqrt{(n^2(n+1)^2 + (n+1)^2 + n^2)/(n^2(n+1)^2)} = (n^2+n+1)/(n(n+1))`

and further expanded into partial fractions as

`(n^2+n+1)/(n(n+1)) = 1 + \frac1{n(n+1)} = 1 + \frac1n - \frac1{n+1}`

Then the sum telescopes:

`S = \left(1 + \frac11 - \frac12\right) + \left(1 + \frac12 - \frac13\right) + \left(1 + \frac13 - \frac14\right) + \cdots + \left(1 + \frac1{1999} - \frac1{2000}\right)`

`S = 1999 + \frac11 - \frac1{2000} = 2000 - \frac1{2000} = \boxed{(3,999,999)/(2000)}`