Is 0! Really the Same as Dividing by Zero in Series Homework?

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Homework Statement



Stumbled onto this picture..

Homework Equations


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The Attempt at a Solution



I see the first term in the series has in the denominator 0! but isn't that the same thing as dividing by 0 or do we treat the first term as just the number 1?
 
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0! is by convention 1. You want n!=n*(n-1)!. So 1!=1 should be 1*0!. Better set 0! to one.
 
e^0=1 and using that formula, we get e^0=\lim_{n\to\infty}\left(\frac{1}{0!}+\frac{0}{1!}+\frac{0}{2!}+...+\frac{0}{n!}\right)=\frac{1}{0!} so you can then conclude that since we have 1=\frac{1}{0!} then 0!=1
 
You can also use the gamma function for a quick way of seeing this.
\Gamma (x) = \int_0 ^{\infty} t^{x-1} e^{-t} dt for x>0. Plugging in x = 1, you see that \Gamma (1) = 1. For integers, the gamma function has the recursion \Gamma (n+1) = n!, so for n = 0 we have 1 = \Gamma (1) = 0!.
 
I little unrelated, but I was going through all of the homework threads to give myself some much needed practice.
But none of them made me smile like this : )
 
Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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