Evaluate the limit by first recognizing the sum as a Riemann Sum for a function defined on $[0,1]$
$$
\lim\limits_{n \to \infty} \sum\limits_{i=1}^{n} \left(\frac{i^5}{n^6}+\frac{i}{n^2}\right).
$$
I understand how to do a Riemann Sum but am not sure how to find f(x) from the limit.
I apologize for any errors in the way I've posted this. This is my first question on here so I don't quite know the swing of things yet.
Thanks everyone for your help!
Best Answer
If you pull out $\frac{1}{n}$ you get $$ \frac{1}{n}\sum_{i = 1}^{n} \left[\left(\frac{i}{n}\right)^5 + \left(\frac{i}{n}\right)\right], $$ which is a Riemann sum for the function $f(x)=x^5+x$ on the interval $[0,1]$. Your limit thus equals $$ \int\limits_0^1(x^5+x)dx = \left[ \frac{x^6}{6} + \frac{x^2}{2} \right]_{x = 0}^{1} = \frac{1}{6} + \frac{1}{2} = \frac{2}{3}. $$