Let 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. Find the power series for the derivative f′(x)f'(x)f′(x).
∑n=1∞xn−12n⋅(n−1)!\sum_{n=1}^{\infty} \frac{x^{n-1}}{2^n \cdot (n-1)!}∑n=1∞2n⋅(n−1)!xn−1
∑n=0∞xn+12n⋅n!\sum_{n=0}^{\infty} \frac{x^{n+1}}{2^n \cdot n!}∑n=0∞2n⋅n!xn+1
∑n=1∞xn2n⋅n!\sum_{n=1}^{\infty} \frac{x^n}{2^n \cdot n!}∑n=1∞2n⋅n!xn
∑n=0∞xn2n⋅(n+1)!\sum_{n=0}^{\infty} \frac{x^n}{2^n \cdot (n+1)!}∑n=0∞2n⋅(n+1)!xn