How it can be shown that:
$$\sum_{k=1}^{2n-1}\frac{\left(-1\right)^{k-1}k}{\binom{2n}{k}}=\frac{n}{n+1}$$
for $1 \le n$
I tried to use this method , but that was not helpful, Also I tried to use the following identity: $$\frac{1}{\binom{2n+1}{k}}+\frac{1}{\binom{2n+1}{k+1}}=\frac{2n+2}{2n+1}\ \frac{1}{ \binom{2n}{k}}$$
and use some telescoping property , but again that did not help me.
Please if it's possible,then do the proof using elementary ways.