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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!

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

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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*}

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  • $\begingroup$ Unless I am mistaken, it suffices to note that $y= x^6$ is increasing, the convexity is not needed. $\endgroup$
    – Martin R
    Jan 20 '20 at 12:19
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    $\begingroup$ @MartinR That's right ... I wish I had said that. I will edit. $\endgroup$ Jan 20 '20 at 12:23

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