I am asked to find the GIF (greatest integer function) of the sum:$$\sum_{i=1}^{1000}\frac{1}{i^{2/3}}$$
I am able to find the lower limit of the sum by using the fact that $$\sum_{i=1}^{1000}\frac{1}{i^{2/3}}$$ is greater than $$\int_{1}^{1000}x^{-2/3}dx$$ But I am unable to find the upper limit which will help me find the GIF. Any help is appreaciated.
Edit : I realized I should add this, the answer is 27. So I basically have to prove that the given sum lies between 27 and 28, of which I have proved 27 is the lower limit.
A strict lower limit is $$\int_1^{1001}x^{-2/3}dx = 3(\sqrt[3]{1001} - 1) > 3(\sqrt[3]{1000} - 1) = 27.$$ A strict upper limit is $$1 + \int_1^{1000}x^{-2/3}dx = 1 + 3(\sqrt[3]{1000} - 1) = 28.$$
So, the answer is $\boxed{27}$.