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minus_twelve_._note [2017/11/25 23:02]
nikolaj
minus_twelve_._note [2019/09/09 22:45]
nikolaj
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 There are theories in math that give meaning to infinite sums, and the standard one, analysis (or, to some reach, calculus) has a million applications for practical applications,​ in particular physics and engineering. The picture above demonstrates the claim There are theories in math that give meaning to infinite sums, and the standard one, analysis (or, to some reach, calculus) has a million applications for practical applications,​ in particular physics and engineering. The picture above demonstrates the claim
  
-$$1+\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+\dots = 2$$+$$1+\dfrac{1}{2}+\dfrac{1}{4}+\dfrac{1}{8}+\dots = 2$$
  
 which can be proven, in analysis. Here's another claim which can be proven, in analysis. Here's another claim
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 So as an example, with $k=3,​a=2,​b=4$ you get the identity So as an example, with $k=3,​a=2,​b=4$ you get the identity
-$$\frac{1}{3}(4^3-2^3)=(2^2+3^2)+\frac{1}{2}(4^2-2^2)-\frac{1}{12}\,​2\,​(4^1-2^1)$$+$$\frac{1}{3}(4^3-2^3)=(2^2+(4-1)^2)+\frac{1}{2}(4^2-2^2)-\frac{1}{12}\,​2\,​(4^1-2^1)$$
 and there are literally infinitely many identities involving $-\frac{1}{12}$ because of this formula. In case you're wondering, both sides of the equation above simplify to $\tfrac{56}{3}$. and there are literally infinitely many identities involving $-\frac{1}{12}$ because of this formula. In case you're wondering, both sides of the equation above simplify to $\tfrac{56}{3}$.
  
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 We've seen this above, actually, in the special case of $z=-\frac{1}{2}$. Indeed, $\frac{1}{1-1/​2}=\frac{1}{1/​2}=2$ and the first formula in this post was We've seen this above, actually, in the special case of $z=-\frac{1}{2}$. Indeed, $\frac{1}{1-1/​2}=\frac{1}{1/​2}=2$ and the first formula in this post was
  
-$$1+\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+\dots = 2$$+$$1+\dfrac{1}{2}+\dfrac{1}{4}+\dfrac{1}{8}+\dots = 2$$
  
 The smooth analogous to the sum with $z$ is The smooth analogous to the sum with $z$ is
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