Can a power series have an interval of convergence of (0,1] or (0,∞)?

[maxpowers_00]

the question is can you come up with a power series whose interval of convergance is the interval (0,1] that is 0 < x < = 1 ? how about (0,infinity)? Give an explicit series of explain why you can't.

The first part of this question where they ask if a series can have an interval of (0,1). I don't think such a series can exist becaue when x=0 doesn't the series always converge, there for the series would have to be eqal to 0. so it can have an interval of [0,1) but not (0,1). but that would also mean a series can't have an interval of (0,infinity).

i just wan to know if it i am going the right way in my thinking and if, not if some one could point me in the right direction.

thanks

 

[Hurkyl]

Power series don't have to be in x; they can be in (x - a) for some a...

 

[M98Ranger]

Your thinking is on the right track. It is not possible for a power series to have an interval of convergence of (0,1] because as you stated, when x=0 the series will always converge to 0. This means that the series would have to be equal to 0 for all values of x in the interval (0,1), which contradicts the definition of a power series.

Similarly, a power series cannot have an interval of convergence of (0,∞) because as x approaches infinity, the series will either diverge or converge to a non-zero value. This means that the series cannot be equal to 0 for all values of x in the interval (0,∞), again contradicting the definition of a power series.

In order for a power series to have an interval of convergence of (0,1), the series would have to converge at x=1 but diverge at x=0. This is not possible for a power series, as it must either converge or diverge for all values of x within its interval of convergence.

Therefore, it is not possible to come up with an explicit power series that has an interval of convergence of (0,1] or (0,∞). The interval of convergence for a power series is determined by the values of x for which the series converges, and it cannot be arbitrarily chosen.