For any n and p≥0 give an argument that the following is true: ∫n0xpdx≤1+2p+3p+⋯+np≤∫n+10xpdx
I’m having trouble even beginning this question. My first thought it to somehow meld this with the squeeze theorem, but, again, am not sure how to begin and show any real work. Any insight is very much appreciated.
Write the integral as the sum
Now, since p>0, xp is an increasing function for x>0. Thus mp<(m+1)p for all m>0. Then, we have
We also have
Using similar reasoning, we see that
Finally, putting it all together reveals