Let f(x)=∑n=0∞xnn!f(x) = \sum_{n=0}^{\infty} \frac{x^n}{n!}f(x)=∑n=0∞n!xn. What is the power series for g(x)=xf(x)g(x) = x f(x)g(x)=xf(x)?
∑n=0∞xn+1n!\sum_{n=0}^{\infty} \frac{x^{n+1}}{n!}∑n=0∞n!xn+1
∑n=1∞xn(n−1)!\sum_{n=1}^{\infty} \frac{x^n}{(n-1)!}∑n=1∞(n−1)!xn
∑n=0∞xnn!\sum_{n=0}^{\infty} \frac{x^n}{n!}∑n=0∞n!xn
∑n=0∞xn+1(n+1)!\sum_{n=0}^{\infty} \frac{x^{n+1}}{(n+1)!}∑n=0∞(n+1)!xn+1