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Here is a funny exercise $$\sin(x - y) \sin(x + y) = (\sin x - \sin y)(\sin x + \sin y).$$ (If you prove it don't publish it here please). Do you have similar examples?

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Maybe a moderator should put the zeta ones together since there are three already? –  anon Nov 3 '10 at 22:29
Perhaps this should be a community wiki question. –  Nuno Nov 3 '10 at 22:31
This is related. –  J. M. Nov 3 '10 at 22:35
I have tripped up many calculus students with this one: $log(1+2+3)=log1+log2+log3$. I am evil... –  user641 Dec 8 '12 at 1:23
@SteveD If only we could find an odd example... –  peoplepower Jan 13 '13 at 0:31

64 Answers 64

If we define $P$ as the infinite lower triangular matrix where $P_{i,j} = \binom{i}{j}$ (we can call it the Pascal Matrix), then $$P^k_{i,j} = \binom{i}{j}k^{i-j}$$

where $P^k_{i,j}$ is the element of $P^k$ in the position $i,j.$

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Voronoi summation formula:

$\sum \limits_{n=1}^{\infty}d(n)(\frac{x}{n})^{1/2}\{Y_1(4\pi \sqrt{nx})+\frac{2}{\pi}K_1(4\pi \sqrt{nx})\}+x \log x +(2 \gamma-1)x +\frac{1}{4}=\sum \limits _{n\leq x}'d(n)$

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$$ \int_{-\infty}^{\infty}{\sin\left(x\right) \over x}\,{\rm d}x = \pi\int_{-1}^{1}\delta\left(k\right)\,{\rm d}k $$

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Here is an Asian kid paradox - perhaps they will understand (apologies for not being strictly mathematical). If $$Study=No \;Fail$$ and $$No \; Study=Fail$$ then $$Study+No\;Study=Fail+No\;Fail$$ $$\implies (1+No)Study=(1+No)Fail$$ Cancelling gives $$Study=Fail$$ Isn't that weird???

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$1+No$ is 0 in both sides. You can't cancel zero... –  CODE Jun 4 '13 at 16:47

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