Prove the following Combinatorial Identity:

Question

I actually would like to try and figure out a proof for myself, but I would like to ask if anyone could provide me a hint to successfully proving the following identity:

$$\sum_{k = 0}^n k \binom{n+1}{k+1} \left(\frac{1}{n} \right)^{k+1} = 1.$$

I tried using the Binomial Theorem for $(1 + \frac{1}{n})^n$ and Pascal's Identity. Is there anything else that I could try? Thank you for your help!

Answer 1

Hope this helps. Let me know if there's something you don't understand.