Proving that $\sum_{k=0}^{n}\frac{(-1)^k}{{n\choose k}}=[1+(-1)^n] \frac{n+1}{n+2}.$ Various examinations ask students to prove that $$\frac{1}{{n \choose k}}=(n+1) \int_{0}^{1} x^k (1-x)^{n-k} dx ~~~~~(1)$$ by evaluating  the integral $\int_{0}^{1}
(tx+1-x)^n~dt$ two ways. When I came across (1), I could prove that $$\sum_{k=0}^{n}\frac{(-1)^k}{{n\choose k}}=[1+(-1)^n] \frac{n+1}{n+2}.~~~~(2)$$ as below:  $$S_n=\frac{1}{n+1}\sum_{k=0}^{n}\frac{(-1)^k}{{n\choose k}}=\int_{0}^1 \sum_{k=0}^{n} (-1)^k  x^k (1-x)^{n-k} dx =  \int_{0}^{1}  (1-x)^n \sum_{k=0}^{n}\left(\frac{-x}{1-x} \right)^k dx =\int_{0}^{1}(1-x)^n \frac{ \left( \frac{-x}{1-x}\right)^{n+1}-1}{\left(\frac{-x}{1-x}\right)-1}dx$$
$$=\int_{0}^{1}[(1-x)^{n+1}-(-x)^{n+1}] dx =\frac{1+(-1)^n}{n+2}.~~~~(3)$$
Therefore, $$S_{2m+1}=0 ~~~~(4) ~~~ \mbox{and}~~~ S_{2m}= \frac{2m+1}{m+1}.~~~~(5)$$
Now the question is: Can one prove (2) in some other way(s)? 
 A: The general formula is: 

$$\sum\limits_{k=0}^n\frac{(-1)^k}{\binom x k} = \left(1+\frac{(-1)^n}{\binom {x+1} {n+1}}\right)\frac{x+1}{x+2}$$

Value range: $\enspace n\in\mathbb{N}_0~,~~ x\in\mathbb{C}\setminus\{n-k|k\in\mathbb{N}\}$ 
The proof by induction with respect to $~n~$ is based on the following equation:
$$\frac{1}{\binom x {n+1}}\frac{x+2}{x+1} = \frac{1}{\binom {x+1} {n+1}} + \frac{1}{\binom {x+1} {n+2}}$$ 
A: I think that the intuition about your result can be the following:
$f(n,k):=\int_0^1 x^k(1-x)^{n-k}>0$
and you can observe that 
$1=1^n=((1-x)+x)^n=\sum_{k=0}^n\binom{n}{k}(1-x)^{n-k}x^k$
so if you integrate the two members with respect to $x$ you get that 
$\int_0^11dx=1=$
$\int_0^1 (\sum_{k=0}^n\binom{n}{k}(1-x)^{n-k}x^k)dx=$
$=\sum_{k=0}^n\binom{n}{k}\int_0^1x^n(1-x)^{n-k}dx$
so 
$1=\sum_{k=0}^n\binom{n}{k}f(n,k)$
and you can observe that the identity is satisfied when 
$f(n,k)=\frac{1}{\binom{n}{k}}\frac{1}{n+1}$
So we want prove by induction on $k>1$ for all fixed $n\geq k$ that $f(n,k)=\frac{1}{\binom{n}{k}}\frac{1}{n+1}$
For $k=0$ you have that 
$f(n,0)=\int_0^1x^0(1-x)^{n-0}dx=-\frac{1}{n+1}[(1-x)^{n+1}]|_0^1=\frac{1}{n+1}$
Now we can hypothesize that the sentence is true for some $k-1$ and we want prove that it is true for $k$:
$f(n,k)=\int_0^1x^k(1-x)^{n-k}dx=$
$-\frac{1}{n-k+1}\int_0^1x^kD((1-x)^{n-k+1})dx=$
$=\frac{1}{n-k+1}k\int_0^1x^{k-1}(1-x)^{n-(k-1)}dx=$
$=\frac{k}{n-k+1}f(n,k-1)=$
$\frac{1}{\binom{n}{k-1}}\frac{1}{n+1} \frac{k}{n-k+1 }=$
$=\frac{(k-1)!k(n-(k-1))!}{n!}\frac{1}{(n+1)(n-(k-1))}=$
$= \frac{1}{\binom{n}{k}}\frac{1}{n+1}$
A: $\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
 \newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
 \newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
 \newcommand{\dd}{\mathrm{d}}
 \newcommand{\ds}[1]{\displaystyle{#1}}
 \newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
 \newcommand{\ic}{\mathrm{i}}
 \newcommand{\mc}[1]{\mathcal{#1}}
 \newcommand{\mrm}[1]{\mathrm{#1}}
 \newcommand{\pars}[1]{\left(\,{#1}\,\right)}
 \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}}
 \newcommand{\root}[2][]{\,\sqrt[#1]{\,{#2}\,}\,}
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 \newcommand{\verts}[1]{\left\vert\,{#1}\,\right\vert}$
$\ds{\bbox[15px,#ffd]{\sum_{k = 0}^{n}{\pars{-1}^{k} \over {n\choose k}} =
\bracks{1 + \pars{-1}^{n}}{n + 1 \over n + 2}}:\ {\Large ?}}$

\begin{align}
\sum_{k = 0}^{n}{\pars{-1}^{k} \over {n\choose k}} & =
\sum_{k = 0}^{n}\pars{-1}^{k}\,{k!\,\pars{n - k}! \over n!}
\\[5mm] & =
\pars{n + 1}\sum_{k = 0}^{n}\pars{-1}^{k}\,
{\Gamma\pars{k + 1}\Gamma\pars{n - k + 1} \over \Gamma\pars{n + 2}}
\\[5mm] & =
\pars{n + 1}\sum_{k = 0}^{n}\pars{-1}^{k}\,
\int_{0}^{1}t^{k}\pars{1 - t}^{n - k}\,\dd t
\\[5mm] & =
\pars{n + 1}\int_{0}^{1}\pars{1 - t}^{n}
\sum_{k = 0}^{n}\pars{-\,{t \over 1 - t}}^{k}\,\dd t
\\[5mm] & =
\pars{n + 1}\int_{0}^{1}\pars{1 - t}^{n}\,
{\bracks{-t/\pars{1 - t}}^{n + 1} - 1 \over
-t/\pars{1 - t} - 1}\,\dd t
\\[5mm] & =
\pars{n + 1}\int_{0}^{1}\pars{1 - t}^{n + 1}\,
\bracks{1 - \pars{-\,{t \over 1 - t}}^{n + 1}}\,\dd t
\\[5mm] & =
\pars{n + 1}\int_{0}^{1}
\bracks{\pars{1 - t}^{n + 1} + \pars{-1}^{n}\, t^{n + 1}}\,\dd t
\\[5mm] & =
\pars{n + 1}
\bracks{-\,{\pars{1 - t}^{n + 2} \over n + 2}
+ \pars{-1}^{n}\, {t^{n + 2} \over n + 2}}_{\ 0}^{\ 1}
\\[5mm] & =
\pars{n + 1}
\bracks{{\pars{-1}^{n} \over n + 2} + {1 \over n + 2}}
\\[5mm] & =
\bbx{\bracks{1 + \pars{-1}^{n}}{n + 1 \over n + 2}} \\ &
\end{align}
A: A telescoping approach:

We obtain
  \begin{align*}
\color{blue}{\sum_{k=0}^n\frac{(-1)^k}{\binom{n}{k}}}
&=\sum_{k=0}^n(-1)^k\frac{k!(n-k)!}{n!}\\
&=\sum_{k=0}^n(-1)^k\frac{k!(n-k)!}{n!}\cdot\frac{(n-(k-1))+(k+1)}{n+2}\\
&=\frac{n+1}{n+2}\sum_{k=0}^n(-1)^k\frac{k!(n-(k-1))!+(k+1)!(n-k)!}{(n+1)!}\\
&=\frac{n+1}{n+2}\sum_{k=0}^n\left((-1)^k\frac{1}{\binom{n+1}{k}}-(-1)^{k+1}\frac{1}{\binom{n+1}{k+1}}\right)\\
&\,\,\color{blue}{=\frac{n+1}{n+2}\left(1+(-1)^n\right)}
\end{align*}
and the claim follows.

