What is the sum of 1/sqrt(n)?

just came across a question
What is the integral value of 1+1/sqrt(2)+1/sqrt(3)+1/sqrt(4)+.....1/sqrt(1000?
i am curious about the general formula till the nth term. This is what i could dig out in a limited timeline:

What's the formula for

n
SUM [1/sqrt(i)]?
i=1

The sum lies
between 2*sqrt(n)-2 and 2*sqrt(n). That is because what you have is
bounded above by

INT [1/sqrt(t) dt] from 0 to n

and below by

INT [1/sqrt(t) dt] from 1 to n

which have the values as indicated.

The formula for the sum is a bit difficult. It is:
n
SUM k^(-1/2) = 2*n^(1/2) - 3/2 + n^(-1/2)/2 -
k=1
n
(1/2)*INT (x-[x]-1/2)*x^(-3/2) dx)
1

Here [x] means the greatest integer less than x. This can be proved by Mathematical Induction.