Infinite sum with factorial in denominator, exponential function in numerator, multiplied with monomial

Question

In practicing for actuarial exam P I came across a problem where I needed
the value of the sum, $$\sum_{n=1}^\infty \frac{n \cdot(1.5^n)}{n!}$$ but I don't know where to begin approaching this, the extra n in the
numerator has really thrown me off

Answer 1

$$\sum_{n=1}^\infty\frac{n(1.5)^n}{n!}=1.5\sum_{n=1}^\infty\frac{1.5^{n-1}}{(n-1)!}=1.5\sum_{n=0}^\infty\frac{1.5^n}{n!}=1.5\times e^{1.5}.$$

Answer 2

Consider $$f(x)=\sum_{n=1}^{\infty} \frac{nx^n}{n!}=x+x^2+\frac{x^3}{2!}+\frac{x^4}{3!}+\cdots=$$ $$x\left(1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots \right)=xe^x.$$ Now plug $x=1.5$: $$f(1.5)=1.5e^{1.5}.$$