Evaluate: $C_0+\frac{C_1}2+\frac{C_2}3+\cdots\frac{C_n}{n+1}$

Question

Evaluate: $C_0+\frac{C_1}2+\frac{C_2}3+\cdots\frac{C_n}{n+1}$.Where $C_k$ denote the usual binomial coefficient.

MY TRY: I solved it and got $\frac{2^{n+1}}{n+1}$,but it's ans. is given as $\frac{2^{n+1}-1}{n+1}$.Thank you.

Answer 1

HINT:

$$(1+x)^n=\sum_{r=0}^n\binom nr x^r$$

$$\int_0^1(1+x)^n=\sum_{r=0}^n\binom nr\int_0^1x^r$$

Answer 2

$$\displaystyle\dfrac{\binom nr}{r+1}=\cdots=\dfrac{\binom{n+1}{r+1}}{n+1}$$

$$\displaystyle\implies\sum_{r=0}^n\dfrac{\binom nr}{r+1}=\dfrac1{n+1}\sum_{r=0}^n\binom{n+1}{r+1}$$ Now, $$\displaystyle\sum_{r=0}^n\binom{n+1}{r+1}=-\binom{n+1}0+\sum_{r=-1}^n\binom{n+1}{r+1}=-1+(1+1)^{n+1}$$