I have a series, $1^3 + 2^3 + 3^3 ... n^3$, and I want to find the upper and lower bound of this series using integrals. I know that for a series that is decreasing (such as $\frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{2^2} ... \frac{1}{n^2}$), the bounds would be as follows:
$ f(x) = \frac{1}{x^2} \\ f(2) + f(3) + f(4)...f(n) \leq \int_1^n f(x)\,\mathrm{d}x \leq f(1) + f(2) + f(3)...f(n-1) \\ 1 \leq \sum_{n=0}^\infty \frac{1}{n^2} \leq 2 $
For a function that is increasing, like the original series, what would the bounds be? The same method wouldn't work because the function is increasing, but I can't seem to figure out the correct ones.
Thanks!