Wrong result for very simple series with Floor

Questions
1

On Mathematica 8 the following sum

Sum[Floor[2/2^p], {p, 1, Infinity}]

evaluates to 0, which is obviously wrong.

Sum[Floor[2/2^p], {p, 1, 10}]

evaluates correctly to 1.

2

Seems like a bug to me.

Sum[Floor[2^-p], {p, 0, Infinity}]

(* 0 *)

Sum[Floor[2^-p], {p, #, Infinity}] & /@ {-1, 0, 1}

(* {3, 0, 0} *)

Strangely enough, this works:

Sum[Floor[2^-p], {p, {-1, 0, 1}, Infinity}]

(* {3, 1, 0} *)
3

Thank you for pointing this out.

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