Consider the power series f(x)=∑n=0∞xn2n⋅n!f(x) = \sum_{n=0}^{\infty} \frac{x^n}{2^n \cdot n!}f(x)=∑n=0∞2n⋅n!xn. Which of the following is true?
f(x)=ex/2f(x) = e^{x/2}f(x)=ex/2 and the series converges for all x∈Rx \in \mathbb{R}x∈R
f(x)=ln(2)+ex/2f(x) = \ln(2) + e^{x/2}f(x)=ln(2)+ex/2 with radius R=∞R = \inftyR=∞
f(x)=ex/2f(x) = e^{x/2}f(x)=ex/2 and the series converges only for ∣x∣<2|x| < 2∣x∣<2
f′(x)=∑n=0∞xn2n+1⋅n!f'(x) = \sum_{n=0}^{\infty} \frac{x^n}{2^{n+1} \cdot n!}f′(x)=∑n=0∞2n+1⋅n!xn and R=∞R = \inftyR=∞