# I don't recognize this limit of Riemann sum

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• mcastillo356
In summary, we discussed the limit of Riemann Sum and the definition of the limit of the General Riemann Sum. The doubts were regarding the expression for the limit and how to bridge the gap. The key is to be specific and provide a specific example rather than a general definition. For instance, the difference between a definition of weather forecast and an actual forecast.
mcastillo356
Gold Member
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?

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!

PeroK said:
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.

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