Integer part of fourth root reciprocal sums

A quick to state puzzle:

If

$$S = 1 + \frac{1}{\sqrt[4]{2}} + \frac{1}{\sqrt[4]{3}} + \dots + \frac{1}{\sqrt[4]{10000}}$$

what is the integer part of $S$?

i.e. what is the integer part of

$$\sum_{k=1}^{10000} \frac{1}{k^{1/4}}$$


(integer part of $x$ = largest integer $\le x$)

[Solution]