Can you cook up a power series whose interval of convergence is the interval (0,1]?How about (0, infinity)? Give an explicit series or explain
why you can't.
for the first consder the power series representation of ln(x+1)
If your studies haven't got you this far look in the taylor series section.
For the second I'll give you a hint -- consider the radius of convergence
for a power series (0,inf) where would the power series be centered?