Why this is wrong let, n be a positive integer.Then,${C_r}=$$ {n} \choose {r}$.Now evaluate $ {C_0}- {C_1}/2+ {C_2}/3+.....+ (-1)^n {C_n}/(n+1)$
I expand $(1-x)^n$ and integrating both side and putting $x=1$ the required series comes.But it gets 0.But answer is $1/n+1$

Another way:
$$\dfrac{\binom nr}{r+1}=\cdots\dfrac{\binom{n+1}{r+1}}{n+1}$$
$$\sum_{r=0}^n\dfrac{(-1)^r\binom nr}{r+1}=\dfrac1{n+1}\sum_{r=0}^n\binom{n+1}{r+1}$$
Now $$\sum_{r=0}^n(-1)^r\binom{n+1}{r+1}=\binom{n+1}0-(1-1)^{n+1}$$
