Question

Question is from definite integral.

Answer 1

The value of the sum is:
`$ (\pi)/(4) $`.

Explanation:

Given:
`$ \lim_(n \to \infty) \displaystyle \sum_(k = 1)^ {n} \bigg ( (n)/(n^2 + k^2) \bigg )`.

Taking
`$ n^2 $` common outside from the denominator, we have:

`$ \bigg ( (n)/((n^2)(1 + ((k^2)/(n^2))) \bigg )`

`\implies(1)/(n) .\frac{1}{1 +\big (  \frac  {k^2}{n^2}\big )}`

We have the following theorem.

If
`$f $` is integrable on [0, 1] then
`$ \int _(0)^(1) f(x) dx = \lim_(n \to \infty) (1)/(n) \displaystyle \sum_(r = 1)^(n) f( (x)/(n))    $`.

Now, let
`$ (k)/(n) = x $`

`$ \implies dx = (1)/(n) $`.

Therefore the summation becomes

`$ = \int_(0)^(1) \bigg ( (1)/(1 + x^2) \bigg )`

`$ \implies \tan^(-1)(x)|_(0)^(1) $`

`$ \implies tan^(-1)(1) - tan^(-1)(0) $`

`$ = (\pi)/(4)  - 0 $`

`$ = (\pi)/(4) $`

Hence, the sum is
`$ \pi $`/4.