Compute $1^{1/3}+2^{1/3}.........7999^{1/3}$

Question

How to compute $$1^{1/3}+2^{1/3}.........7999^{1/3}$$

Answer 1

There is not a closed form for that sum, we can approximate the sum by integral estimation as for example

$$\sum_{k=1}^{7999} x^{1/3}\approx \int_0^{8000} x^{1/3} dx=\left[\frac34 x^{4/3}\right]_0^{8000}= 120\,000$$