Find $\sum\limits_{r=0}^n(-1)^r\binom{n}{r}^{-1}$ for $n$ even 
If $n$ is an even natural number, then find 
  $$\sum_{r=0}^n \left(\frac{(-1)^r}{\binom{n}{r}}\right)$$ 

I tried to solve the question using conventional method, by trying to use calculus, but I don't think that would be applicable here, because no binomial expansion (as far I as know) can give a coefficient in denominator hence I got stuck at it.

Using the beta function, Trif gives a proof in his paper 
COMBINATORIAL SUMS AND SERIES INVOLVING
INVERSES OF BINOMIAL COEFFICIENTS 
to a slightly more general statement:

You question corresponds to the case when $m=0$.

See also the paper Alternating Sums of the Reciprocals
of Binomial Coefficients for a more general discussion about the sum
$$
T_n^{(l,m)}=\sum_{k=0}^n(-1)^kk^l\binom{m+n}{m+k}^{-1}.
$$
