_The outline of this proof follows that which you need to [prove](Representation%20of%20𝛿(y-x).md#Proving%20that%20a%20function%20converges%20to%20𝛿%20y-x)_that as $n$ goes to infinity, that a particular function gives right to a [Dirac delta function.](Dirac%20delta%20function.md)_ ^97ece9 Showing that this representation is normalized to 1 requires us to evaluate a [Gaussian integral](Gaussian%20integral.md). If we do so we find that $\int_{-\infty}^{\infty}dx\frac{n}{\sqrt{\pi}}e^{-n^2(y-x)^2}=\frac{n}{\sqrt{\pi}}\bigg(\frac{\sqrt{\pi}}{|n|}\bigg)=1.$ Thus, .md#^9ec0e7) ^153546 In order to show #^58f03f) we expand $f(x)$ to $f(x)=f(y)+f'(y)(x-y)+\frac{1}{2}f''(y)(x-y)^2+...,$ plug in the [Guassian representation of $\delta(y-x)$,](Gaussian%20representation%20of%20𝛿(y-x)#^5e8f22) and we move the limit inside of the integral by invoking [Lebesgue's dominated convergence theorem.](Lebesgue's%20dominated%20convergence%20theorem.md) When then evaluate it by [substituting variables](Analysis%20(index).md#Substitution%20rules) such that $x\rightarrow u$ and $u = nx$ : $\int_{-\infty}^{\infty} du \lim_{n\rightarrow\infty} f(u/n) \delta_n(y-u/n)=$$\int_{-\infty}^{\infty} du \lim_{n\rightarrow\infty}\bigg(f(y)+f'(y)\bigg(\frac{u}{n}-y\bigg)+\frac{1}{2}f''(y)\bigg(\frac{u}{n}-y\bigg)^2+...\bigg)\frac{n}{\sqrt{\pi}}e^{-(y-u)^2}$ $=f(y)$ ^93c088 %%This last step with the expansion needs to be checked and rewritten given that it seems that the way this type of problem can be solved means that this expansion step applies generally for all of these and we thus only need to do it once in the outline of the proof and then not repeat it here in these convergence proofs.%%  <font size="2"> We can visually see how as $n$ increases, the [[Gaussian function]] centered at $y=0$ converges to $\delta(x)$ at its mean. Notice that this corresponds to a sequence of normal distributions where the standard deviation is $\sigma=1/(\sqrt{2}\sigma)$ with a mean of $\mu=0$.</font> ^324bcd #MathematicalFoundations/Analysis/GeneralizedFunctions #MathematicalFoundations/Analysis/Functions #Proofs