Final answer:
A power series with an interval of convergence of [0, ∞) is the Maclaurin series for the exponential function e^x, which converges for all x in the real numbers. Its form is Σ(x^n)/n!, and because it converges over the entire real line, it satisfies the requirement of the question.
Step-by-step explanation:
The question at hand is whether there exists a power series whose interval of convergence is [0, ∞) or to explain why no such series is possible. A power series has the form Σa_n(x-c)^n, where 'c' is the center of the series, and the a_n's are the coefficients. To find a power series with the desired interval of convergence, we consider functions that are defined and convergent for all non-negative real numbers.
One such function is the exponential function e^x, which has a power series expansion that converges for all x in the real numbers. The Maclaurin series (a Taylor series centered at 0) for e^x is given by Σ(x^n)/n!, with n ranging from 0 to infinity. This series has an interval of convergence of (-∞, ∞), which certainly includes [0, ∞). Therefore, the Maclaurin series for e^x is one example of a power series whose interval of convergence includes the desired range. It's also essential to note that while this power series converges on the entire real number line, some series may only converge on a subset of it. The radius of convergence dictates the interval over which a power series will converge, determined by the ratio test or other convergence tests.