DREAM1

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1 = \sum_{k=0}^\infty \frac{1}{k!} \sum_{j=0}^\infty \frac{k!}{j!(k-j)!} (i\pi)^k\,
= \frac{1}{2\pi i} \oint_{|z|=2} \frac{1}{z} \sum_{k=0}^\infty \frac{(-1)^k z^{2k}}{(2k)!} dz = 1\,





The above is just a selection of the equations from below:

Binomial Theorem: (2\pi i)^k = \sum_{j=0}^\infty {k \choose j} (i\pi)^k\,

Complex 1 and Taylor expansion of the exponential, with k!=\Gamma(k+1)\,:

1 = e^{i 2\pi} = \sum_{k=0}^\infty \frac{(i 2\pi)^k}{\Gamma(k+1)}\,

And from Contour Integrals:

\frac{1}{i 2\pi} \oint_{|z|=2} \frac{\cos z}{z-0} dz = \frac{i 2\pi}{i 2\pi} = 1\,

And the Taylor expansion:

\cos(x) = \sum_{k=0}^\infty \frac{(-1)^k x^{2k}}{(2k)!}\,


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