Computing an integral from a limit: $\lim_{n\to\infty}\sum_{k=1}^n\frac{n+k}{n^2}$

Question

I want to express the following limit

$$\lim_{n\to\infty}\sum_{k=1}^n\frac{n+k}{n^2}$$

as an integral. And I'm not sure how to compute this integral by interpreting it as the area of a known domain.

EDIT: I tried to use definiton of definite integral's formula but it does not resemble the right hand side to identify the $a$,$b$ and $n$ terms.

Answer 1

You may use a Riemann sum giving, as $n \to \infty$, $$ \sum_{k=1}^n\frac{n+k}{n^2}=\frac1{n}\sum_{k=0}^{n}\left(1+\frac{k}{n} \right) \to \int_0^1(1+x)\:dx=\color{red}{\frac32}. $$

One may recall that, as $n \to \infty$,

$$ \frac1{n}\sum_{k=0}^{n}f\left(\frac{k}{n} \right) \to \int_0^1f(x)\:dx $$

for any Riemann integrable function $f$ on $[0,1]$.

Answer 2

HINT:

$$\lim_{n\to\infty}\sum_{k=1}^n\frac{n+k}{n^2}=\lim_{n\to\infty}\sum_{k=1}^n\frac{1+\frac{k}{n}}{n}=\lim_{n\to\infty}\frac{1}{2}\left(\frac{1}{n}+3\right)=\frac{1}{2}\left(3\right)=\frac{3}{2}$$