Compute $\sum\limits_{k=1}^{9999}\frac1{2\sqrt k+1}$

Question

I'm in trouble when trying to calculate the following sum:

$$\frac1{2\sqrt 1+1}+\frac1{2\sqrt 2+1}+\frac1{2\sqrt 3+1}+\ldots+\frac1{2\sqrt {9999}+1}$$

Since there isn't any formula to calculate the series$\sum\frac1{n^{1/2}}$ and there are square root numbers in the sum, how do you solve this?

Please note that this problem was given in a tournament in which I participated, so it MUST be solvable.

Answer 1

You can get quite a good approximation by evaluating $$\int_1^{9999}\frac{1}{2\sqrt{x}+1}dx=\sqrt{9999}-1-\frac 12\ln\left(\frac{2\sqrt{9999}+1}{3}\right)\simeq96.89267844$$