# Upper and Lower bound of a finite sum

Find upper and lower bound for the following finite sum:

$$\frac{1}{1} + \frac{1}{2^3} + \frac{1}{3^3} + ··· + \frac{1}{n^3}$$

My attempt is:

Using the integral test:

we know that $$\frac{1}{1} + \frac{1}{2^3} + \frac{1}{3^3} + ··· + \frac{1}{n^3}$$ = $$\sum_{i=1}^n 1/i^3 = \int_1^n1/i^3di = \int_1^n1/x^3dx = -1/2n^2 + 1/2$$

But now I'm stuck. How can this give the lower and upper bouunds?