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?


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