The sum of series $\frac{1}{1!}+\frac{1+2}{2!}+\frac{1+2+3}{3!}+\ldots+\infty$ equals

Question

Professor PC Thomas Classes · Answer

$T_n = \frac{n(n+1)/2}{n!} = \frac{n+1}{2(n-1)!}$. Sum resolves to combination of Taylor series for $e$, yielding $\frac{3e}{2}$.