Mazeraiʻs Criterion
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Mazeraiʻs Criterion
This note deals with mazeraiʻs criterion which is all about whether the body is the “right fit” for a dead-zone. A dead-zone is a place or location with a lot of lead to block cosmic experience. It basically makes you feel alone rather than one with the cosmos.
Mazeraiʻs Criterion (Leviathan)
We want to show that integration for $f(x) = e^x$ is not the same via partial integration of the parts of the Taylor Series. So, first, since we know
\[\frac{d}{dx}e^x = e^x\]we can immediately infer that…
\[\int e^x dx = e^x + C\]so we know the solution. In terms of the infinite Taylor Series, we have
\[f(x) = e^x = \sum_{i=0}^\infty \frac{x^i}{i!} = 1 + x + \frac{x^2}{2} + \frac{x^3}{6} + \dots\]so, what we end up with when we take partial integration as…
\[\int f(x)dx = \int e^x dx = \int \sum_{i = 0}^\infty \frac{x^i}{i!} dx \\ = \sum_{i = 0}^\infty \frac{1}{i!}\int x^i dx = \sum_{i=0}^\infty \frac{1}{i!}\left[\frac{x^{i+1}}{i+1} + C\right] = \sum_{i = 0}^\infty \frac{x^{i+1}}{(i+1)!} + C \\ = e^x - 1 + C = e^x + C^\prime\]So, what the integration with the Taylor Series tells us is that it should be regraded as “partial integration” due to the $C^\prime$ constant. So, when we integrate with respect to a Taylor Series, we call it a “partial-integration.” We know,
\[\int e^x dx \neq \int_{\partial} e^x dx\]for this case! This is as desired. $\blacksquare$
Time heals all wounds!