It is well known that
e=k=1∑∞(k−1)!1=0!1+1!1+2!1+3!1+⋯
This value is also known (see Calvin's problem)
k=1∑∞(k−1)!k=0!1+1!2+2!3+3!4+⋯=x (x is kept secret here so Calvin's problem isn't spoiled.)
So what is
k=1∑∞(k−1)!k2=0!1+1!4+2!9+3!16+⋯?
Notation: ! denotes the factorial notation. For example, 8!=1×2×3×⋯×8. And 0!=1 as always.