The Numerators Diverge Too
Calculus
Level
2
It is well known that
\[ e = \sum_{k=0}^\infty \dfrac{1}{k!} = \dfrac{1}{ 0!} + \dfrac{1}{1!} + \dfrac{1}{2!} + \dfrac{1}{3!} + \cdots \]
What is the value of
\[ \sum_{k=0}^{\infty} \dfrac{ k+1} {k!} = \dfrac{1}{0!} + \dfrac{2}{ 1!} + \dfrac{3}{2!} + \dfrac{ 4}{3!} + \cdots \quad ? \]
\[\] Notation: \(!\) denotes the factorial notation. For example, \(8! = 1\times2\times3\times\cdots\times8 \).
Bonus: Prove this by just manipulating the terms from the first identity.
by
Calvin Lin