<s>[INST] 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?
 [/INST] \begin{eqnarray}
&&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}\\
&=&\sum_{k=0}^n(-1)^{k}\frac{1}{k+1}\binom{n}{k}\\
&=&\sum_{k=0}^n(-1)^{k}\frac{1}{n+1}\binom{n+1}{k+1}\\
&=&\frac{1}{n+1}\sum_{k=1}^{n+1}(-1)^{k}\binom{n+1}{k}\\
&=&-\frac{1}{n+1}\left[\sum_{k=0}^{n+1}(-1)^{k}\binom{n+1}{k}-1\right]\\
&=&\frac1{n+1}
\end{eqnarray}
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