Inequalities involving summation

Question

I have been struggling with the following math problem for quite a while without any real progress:

Let $\ n \in \mathbb N$

Show that $$\frac{127}{7}(n-1)^7 \le \sum_{n=k}^{2n-1} k^6 \le \frac{127}{7}n^7.$$

I would like to know how to approach the problem above and also how to approach these kinds of problems in general since I barely have any previous experience with them. Thanks a lot in advance!

Answer 1

The function $y=x^6$ is increasing, so by considering the area under the curve
\begin{eqnarray*} \int_{n-1}^{2(n-1)} k^6 dk \leq \sum_{k=n}^{2n-1} k^6 \leq \int_{n}^{2n} k^6 dk. \end{eqnarray*}

Answer 2

The right inequality you can prove by induction, for which prove that: $$\frac{127n^7}{7}+(n+1)^6\leq\frac{127(n+1)^7}{7}$$ and