Now, I don't claim to be a Mathematics-overlord (I'm still a freshman ;P ), but I really don't understand the output Mathematica produces. I really think it is simply false. No I don't think I found a bug; I'm pretty positive this is a design choice; but I really don't understand it.

Here we go:

In[1]:= Clear[q, k]

In[2]:= Sum[q^k, {k, Infinity}]

Out[2]= -(q/(-1 + q))

In[3]:= Reduce[
 ForAll[q, Element[q, Reals], 
  Sum[q^k, {k, Infinity}] == -q/(-1 + q)], q, Reals]

Out[3]= True

In[4]:= SumConvergence[q^k, k]

Out[4]= Abs[q] < 1

I'm sorry, but while I can understand the output of the Sum function alone, all of this simply turns false if put under the universal quantifier. For some values of q the series is not convergent. It is especially weird for q=1, as we have division by zero then. Hence it is not true that for all real values of q this sum equals -q/(-1+q).

Could somebody be kind enough and explain this to me? Am I failing to understand something, or am I doing something wrong?

Thanks in advance, Marcin G.