is it possible to have a powerr series that converges with interval of 0 inclusive to inf?
No, there isn't. In his first response Archie was concerned that you "inclusive" referred to the "inf" as well as to 0. Yes, you can include 0 in an interval, no you cannot include "infinity" because that is not a number in the sense being used here. You can have [0, inf). You cannot have [0, inf].
for a power series we cannot have an interval of convergence of the form $\displaystyle [0,\infty )$
the reason is that if a power series converges at $\displaystyle x=a\neq 0$
then it also converges for all $\displaystyle x$ with $\displaystyle |x|<|a|$