# Inequalities involving summation

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!

• Induction might work ? or area under a graph approximated by rectangles Jan 20 '20 at 10:18

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
The function $$y=x^6$$ is increasing, so by considering the area under the curve
$$\begin{eqnarray*} \int_{n-1}^{2(n-1)} k^6 dk \leq \sum_{k=n}^{2n-1} k^6 \leq \int_{n}^{2n} k^6 dk. \end{eqnarray*}$$
• Unless I am mistaken, it suffices to note that $y= x^6$ is increasing, the convexity is not needed. Jan 20 '20 at 12:19