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How can I have mathematica automatically simplify expressions like this

Sum[Exp[-I x k], {x, -Infinity, Infinity}]

using

KroneckerDelta[k]

? Evaluating the above using Simplify produces: Sum does not converge.

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Look at FourierSequenceTransform:

FourierSequenceTransform[1, x, k]

2 π DiracDelta[k]

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