# Binomial theorem relating proof

There is this identity

$$1 -\frac{1}{2}\binom{n}{1}+\frac{1}{3} \binom{n}{2}- \frac{1}{4}\binom{n}{3}+....+(-1)^n \frac{1}{n+1}\binom{n}{n}$$

And we are supposed to prove it using these two identities

$$k\binom{n}{k} = n\binom{n-1}{k-1}$$

and

$$\binom{n}{0} + \binom{n}{1} + \binom{n}{2} +....+ \binom{n}{n} = 2^n$$

I have been working on this problem for a long time. Can you guys help me?

• Your first expression? – user418131 Oct 3 '17 at 19:24
• Is this correct as written? I do not see an equality to prove – TomGrubb Oct 3 '17 at 19:30

• $\pm \frac{1}{n+1}$ ... lol ... I was about to change my answer to agree with yours $\ddot \smile$ – Donald Splutterwit Oct 3 '17 at 19:42
• @J.Chang, use $(1-1)^{n+1}=0$. – xpaul Oct 4 '17 at 13:52