This result surprised me, since the two expressions actually are quite close to one another.
Δ ≈ 1.83 × 10⁻⁴
The Gaussian Sum (2) has no closed formpic.twitter.com/HpVZEySlev
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(1) The limit as a Gaussian's width (1/α) goes to zero is 0 (2) The same limit of the "Gaussian Sum" however is 1pic.twitter.com/3U7I4zsQ1r
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Nope. Improper integral., = 1 too
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But the limit isn't 0 for all values. There's one term of that sum causing a problem!
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Proof or this is false
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Great!
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One can think of the second situation as being a “pixelated” version of the first, where the limiting case is one lit up pixel with area 1
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just take out the n = 0 term in (2) ?
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I strongly agree: the difference lies in the neighbourhood of 0, where the contribution to the sum doesn't vanish in the limit, which makes the result less surprising imho
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