Summation of binomial coefficients upon $(r+2)$

Question

$$\sum^{50}_{r=0}(-1)^r \dfrac{\dbinom {50}r}{r+2}= ?$$

Attempt:

$(1-x)^{50}= \sum (-1)^r \dbinom{50}r x^{n-r}$

Integrating both sides and then placing limits 0 to 1:

$\dfrac{1}{51}= \displaystyle \sum_{r=0}^{50}\dfrac{(-1)^r \dbinom{50}r}{52-r}$

So the answer should be $\dfrac{1}{51}$ But answer given is $1/(51\times 52)$

Where have I gone wrong?

Edit:

I understood my mistake from the comment. Now I have: $\dfrac{1}{51}= \displaystyle \sum_{r=0}^{50}\dfrac{(-1)^r \dbinom{50}r}{51-r}$

Is it possible to complete it from here?

Answer 1

$\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,} \newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace} \newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack} \newcommand{\dd}{\mathrm{d}} \newcommand{\ds}[1]{\displaystyle{#1}} \newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,} \newcommand{\ic}{\mathrm{i}} \newcommand{\mc}[1]{\mathcal{#1}} \newcommand{\mrm}[1]{\mathrm{#1}} \newcommand{\pars}[1]{\left(\,{#1}\,\right)} \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}} \newcommand{\root}[2][]{\,\sqrt[#1]{\,{#2}\,}\,} \newcommand{\totald}[3][]{\frac{\mathrm{d}^{#1} #2}{\mathrm{d} #3^{#1}}} \newcommand{\verts}[1]{\left\vert\,{#1}\,\right\vert}$ \begin{align} \sum_{r = 0}^{50}\pars{-1}^{r}\,{{50 \choose r} \over r + 2} & = \sum_{r = 0}^{50}\pars{-1}^{r}\,{50 \choose r}\int_{0}^{1}t^{r + 1}\,\dd t = \int_{0}^{1}t\sum_{r = 0}^{50}{50 \choose r}\pars{-t}^{r}\,\dd t = \int_{0}^{1}t\,\pars{1 - t}^{50}\,\dd t \\[5mm] & = {\Gamma\pars{2}\Gamma\pars{51} \over \Gamma\pars{53}} = {1!\,50! \over 52!} = {1 \over 52 \times 51} = \bbx{1 \over 2652} \end{align}

Answer 2

You have to integrate not $(1-x)^{50}$ but $x(1-x)^{50}$: $$ (1-x)^{50}= \sum_{r\ge0} (-1)^r \dbinom{50}r x^{r}\Rightarrow\\ x(1-x)^{50}=x\sum_{r\ge0} (-1)^r\dbinom{50}r x^{r}=\sum_{r\ge0} (-1)^r\dbinom{50}r x^{r+1}\Rightarrow\\ \int_0^1 x(1-x)^{50}dx=\left[\sum_{r\ge0} (-1)^r\dbinom{50}r\frac{x^{r+2}}{r+2}\right]_0^1=\sum_{r\ge0} (-1)^r\dbinom{50}r\frac{1}{r+2}. $$

The integral on LHS can be computed by parts: $$ \int_0^1 x(1-x)^{50}dx= \left[-\frac{x(1-x)^{51}}{51}\right]_0^1+\frac{1}{51}\int_0^1 (1-x)^{51}dx=\frac{1}{51\cdot52}. $$

Answer 3

OP's expression is a special case of a formula for the reciprocal binomial coefficient using the Beta function

\begin{align*} \color{blue}{\binom{n}{k}^{-1}}&\color{blue}{=(n+1)\int_0^1 z^k(1-z)^{n-k}\,dz}\\ &=(n+1)\int_0^1z^k\sum_{r=0}^{n-k}\binom{n-k}{r}(-z)^r\,dz\\ &=(n+1)\sum_{r=0}^{n-k}\binom{n-k}{r}(-1)^r\int_0^1z^{k+r}\,dz\\ &\,\,\color{blue}{=(n+1)\sum_{r=0}^{n-k}\binom{n-k}{r}\frac{(-1)^r}{k+r+1}}\tag{1} \end{align*}

We obtain from (1) \begin{align*} \sum_{r=0}^{n-k}\binom{n-k}{r}\frac{(-1)^r}{k+r+1}=\frac{1}{(n+1)\binom{n}{k}}\tag{2} \end{align*}

Putting $n=51$ and $k=1$ in (2) we derive OP's formula \begin{align*} \color{blue}{\sum_{r=0}^{50}\binom{50}{r}\frac{(-1)^r}{r+2}}=\frac{1}{52\binom{51}{1}}\color{blue}{=\frac{1}{52\cdot 51}} \end{align*}