It is well known that
e=k=0∑∞k!1=0!1+1!1+2!1+3!1+⋯
What is the value of
k=0∑∞k!k+1=0!1+1!2+2!3+3!4+⋯?
Notation: ! denotes the factorial notation. For example, 8!=1×2×3×⋯×8.
Bonus: Prove this by just manipulating the terms from the first identity.