I want to find $f(x)$ throught calculus, where $f(x) \equiv\sum_{i=1}^{x} i$
This is my reasoning:
let $ g(x) = x $, and $ \Delta x = \frac{1}{n}$ . Then, by Riemann integration,
$\sum_{i=0}^{x} i \sim \sum_{i=0}^{x} \frac{i}{n}·\frac{1}{n} \implies \lim_{n \to \infty} \sum_{i=0}^{x} \frac{i}{n}·\frac{1}{n} = \int_{0}^{x}x·dx =\frac{x^2}{2}$,
But Gauss found that
$\sum_{i=0}^{x}i = \frac{x^2+x}{2} $.
What part of the reasoning is false or bad developed?
Going backwards, knowing that
$F(x) = \frac{x^2+x}{2} $.
this function is the primitive of $f'(x) = x+\frac{1}{2}$, not just $x$.
Again, what is wrong? I love maths but i havent studied them in a high level, so if anyone could explain me I would be so grateful, thanks!!
$\endgroup$3 Answers
$\begingroup$First, I have no idea what you are using $\sim$ to mean but on the left you have $0 + 1 + 2 + \dots + x$ and on the right you have $\frac{0}{n^2} + \frac{1}{n^2} + \dots + \frac{x}{n^2}$. While we're here: is $x$ an integer? Otherwise you should be more precise in what you mean. I also don't know what you are using the $\implies$ (implies) symbol to mean.
Secondly, if you take this sum $\frac{0}{n^2} + \frac{1}{n^2} + \dots + \frac{x}{n^2}$ and let $n \to \infty$ you get $0$ since each term goes to $0$.
Thirdly, writing $\int_0^x x \;dx$ is confusing because the variable inside the integral should not be the same variable use for the region of integration.
$\endgroup$ 1 $\begingroup$one really can solve some summation problems with integrals, but even your first statement $\Delta x=\frac{1}{n}$ is not valid if you wan a Riemann sum up to x or to a, better to choose a instead of the variable x then one has to choose $\Delta x=\frac{a}{n}$ the sum would then be $\sum_{i=0}^n\frac{ai}{n}*\frac{a}{n}\ne\sum_{i=0}^n i $so your integral does not help you to find this sum
$\endgroup$ $\begingroup$One of the first things--among others--worth considering here is that, what appears to be the intended, infinitesimal element necessary for convergence is not very appropriate for the purposes of that finite series; although that pitfall is understandable.
But, as an addition to the replies so far, the most enlightening approach--short of a course on Real Analysis--might be to see a constructive proof of the method used to reach the correct answer.
So, to that end, here are some (hopefully appropriately elementary) resources:
Wikipedia's article on the Euler-Maclaurin Forumla (and by extension: Faulhaber's Polynomials) seems to contain a constructive proof--apparently due to Apostol's “An Elementary View of Euler’s Summation Formula”. They appear, at a glance, to be fairly approachable.
In addition, also are Notes on Euler-Maclaurin, which looked like good exposition. That in particular, also points to additional references, such as:Apostol (1973) as well as an article on its history; worth noting, also, as an interesting aside, its extension to singular functions
I hope this is helpful.
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