I don't recognize this limit of Riemann sum

[mcastillo356]

TL;DR Summary

I look at the limit, and I look at the definition, and I don't match both concepts, though I should.

Hi, PF, I hope the doubts are going to be vanished in a short while:

This is the limit of Riemann Sum
$\displaystyle\lim_{n\rightarrow{\infty}}\displaystyle\frac{1}{n}\displaystyle\sum_{j=1}^{n}\cos\Big(\displaystyle\frac{j\pi}{2n}\Big)$

And this is the definition of the limit of the General Riemann Sum:
Let $P=\{x_0,x_1,x_2,...,x_n\}$ where $a=x_0<x_1<x_2<\cdots{<x_n=b}$, be a partition of $[a,b]$ having norm $||P||=\mbox{max}_{1\leq i\leq\n}\,Deltax_i$. In each subinterval of $P$ pick a point $c_i$ (called a tag). Let $c=(c_1,c_2,...,c_n)$ denote the set of these tags. The sum $R(f,P,c)=\displaystyle\sum_{i=1}^n\,f(c_i)\Delta{x_i}=f(c_1)\Delta{x_1}+f(c_2)\Delta{x_2}+f(c_3)\Delta{x_3}+\cdot{f(c_n)\Delta{x_n}}$ is called the Riemann sum of $[a,b]$ corresponding to partition $P$ and tags $c$.

Doubts: On the expression $\displaystyle\lim_{n\rightarrow{\infty}}\displaystyle\frac{1}{n}\displaystyle\sum_{j=1}^{n}\cos\Big(\displaystyle\frac{j\pi}{2n}\Big)$, how must I manage to bridge the gap?

 

[PeroK]

The answer is specificity. In this case, it means the difference between a general definition and a specific example.

For example, the definition of a weather forecast is very different from an actual forecast.

 

[mcastillo356]

Hi, PF, @PeroK, thanks a lot!

The answer is specificity. In this case, it means the difference between a general definition and a specific example.

For example, the definition of a weather forecast is very different from an actual forecast.

Greetings.