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4:22 PM
The first fails to evaluate. The second returns a[0]. They're both equivalent. I feel that this would have been asked before, but I can't find a Q&A. It's fairly straightforward to turn the first into the second, so I'm not sure it's worth asking. It would be nicer if there were a robust way to help Sum handle such things.
Sum[Piecewise[{{1, k == 0}}, 0]*a[k], {k, 0, n}]
Sum[KroneckerDelta[k]*a[k], {k, 0, n}]
 

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