# Infinite Sum - Result not correct for all cases?

Evaluating the Sum

Sum[a^i, {i, ∞}]


yields

-(a/(-1 + a))


which obviously only holds true for $\left|a\right|<1$. Why doesn't Mathematica give a limitation for the validity of the solution?

• Sum[a^i, {i, ∞}, GenerateConditions -> True] will return the necessary conditions. – J. M. is computer-less Jul 27 '16 at 8:09
• @J.M. OK, thanks. It seems as if I have to get used to MMA thinking more than I do ;) – DPF Jul 27 '16 at 8:18
• It seems a conspicuous inconsistency to me that Sum has GenerateConditions->False by default , while Integrate has it True ( eg Integrate[a^i, {i, 1, Infinity}] ). – george2079 Jul 27 '16 at 20:40

Thanks to @J. M. this solution gives the answer:

Sum[a^i, {i, ∞}, GenerateConditions -> True]


It returns a ConditionalExpression, in the above case:

ConditionalExpression[-(a/(-1 + a)), Abs[a] < 1]

• In addition: look at SumConvergence[a^i, i]. – J. M. is computer-less Jul 27 '16 at 11:29