Hello! First post, over here, hopefully it'll pique some interest, and awesome answers, but any and all answers are welcome!(adsbygoogle = window.adsbygoogle || []).push({});

So, point is, we were given a guest lecture over how to 'jokingly' find answers to specific problems using methods that appear to not be justified; now, after the lecture I asked a question regarding such, and I was told that such operations are actually justified, what is not is the notation. Anyways, enough storytelling, the problem and 'derivation' is as follows:

Given some function [itex]f(x)[/itex] and defining [tex]\int f = \int_0^x f(t) dt[/tex] we wish to find the function [itex]f(x)[/itex] such that [tex]\int f = f-1[/tex] Is the case. Then, after rearranging, we get:

[tex]1=f-\int f[/tex]

We "factor" and receive:

[tex]1=\left(1-\int\right)f[/tex]

Then, taking the "inverse":

[tex]f=\left(1-\int\right)^{-1}1[/tex]

We, then (somehow, magically, as I see it) "apply" the geometric series to receive:

[tex]f=\left(1+\int+\int^2+\int^3+\cdots\right)1[/tex]

[tex]f=1+\left(\int 1\right)+\left(\int^2 1\right)+\left(\int^3 1\right)+\cdots[/tex]

[tex]f=1+x+\left(\int x\right)+\left(\int^2 x\right)+\cdots[/tex]

[tex]f=1+x+\frac{x^2}{2}+\left(\int \frac{x^2}{2}\right)+\cdots[/tex]

[tex]f=1+x+\frac{x^2}{2}+\frac{x^3}{6}+\cdots[/tex]

[tex]f=e^x[/tex]

Which is, very obviously, the correct answer. Anyways, could anyone explain what isactuallygoing on here? I know that the integrals can be dealt with as operators and such... but the notation confuses absolutely anything I have been able to come up with.

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# Playing with Analytical Operators

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