Determine the Taylor series for f(x)=2xf(x) = 2^xf(x)=2x centered at x=0x=0x=0.
∑n=0∞2nxnn!\sum_{n=0}^{\infty} \frac{2^n x^n}{n!}∑n=0∞n!2nxn
∑n=0∞(ln2)nxnn!\sum_{n=0}^{\infty} \frac{(\ln 2)^n x^n}{n!}∑n=0∞n!(ln2)nxn
∑n=0∞xnn!\sum_{n=0}^{\infty} \frac{x^n}{n!}∑n=0∞n!xn
∑n=0∞xn2nn!\sum_{n=0}^{\infty} \frac{x^n}{2^n n!}∑n=0∞2nn!xn