How to show that $\sum_{n=0}^{N}\frac{(-1)^n}{{(N+n)!}(N-n)!}=\frac{1}{2N!^2}$?

Question

How to show that $\displaystyle\sum_{n=0}^{N}\frac{(-1)^n}{{(N+n)!}(N-n)!}=\frac{1}{2N!^2}$?

How show that $(1)$ is: $$\sum_{n=0}^{N}\frac{(-1)^n{2n \choose n}{N+n \choose N-n}}{[(n+1)(n+1)(n+2)(n+3)\cdots(n+N)]^2}=\frac{1}{2N!^2}\tag1$$

using : $\frac{\Gamma(n+N+1)}{\Gamma(n+1)}=(n+1)(n+2)\cdots(n+N)$

simplify to: $$\frac{{2n \choose n}{N+n \choose N-n}}{[(n+1)(n+1)(n+2)(n+3)\cdots(n+N)]^2}=\frac{1}{(N-n)!(N+n)!}$$

$$\sum_{n=0}^{N}\frac{(-1)^n}{{(N+n)!}(N-n)!}\tag2$$

Answer 1

Let $ N $ be a positive integer.

Observe that : $$ \left(\forall n\in\mathbb{N}\right),\ \frac{1}{\binom{N+n}{n}}=\left(n+N+1\right)\int_{0}^{1}{x^{n}\left(1-x\right)^{N}\,\mathrm{d}x} $$

Thus, \begin{aligned} \sum_{n=0}^{N}{\frac{\left(-1\right)^{n}\left(N!\right)^{2}}{\left(N+n\right)!\left(N-n\right)!}}&=\sum_{n=0}^{N}{\frac{\left(-1\right)^{n}\binom{N}{n}}{\binom{N+n}{n}}}\\&=\sum_{n=0}^{N}{\left(-1\right)^{n}\binom{N}{n}\left(n+N+1\right)\int_{0}^{1}{x^{n}\left(1-x\right)^{N}\,\mathrm{d}x}}\\ &=\int_{0}^{1}{\left(1-x\right)^{N}\sum_{n=1}^{N}{\left(-1\right)^{n}n\binom{N}{n}x^{n}}\,\mathrm{d}x}+\left(N+1\right)\int_{0}^{1}{\left(1-x\right)^{N}\sum_{n=0}^{N}{\left(-1\right)^{n}\binom{N}{n}x^{n}}\,\mathrm{d}x}\\ &=N\int_{0}^{1}{\left(1-x\right)^{N}\sum_{n=1}^{N}{\left(-1\right)^{n}\binom{N-1}{n-1}x^{n}}\,\mathrm{d}x}+\left(N+1\right)\int_{0}^{1}{\left(1-x\right)^{2N}\,\mathrm{d}x}\\ &=-N\int_{0}^{1}{x\left(1-x\right)^{N}\sum_{n=0}^{N-1}{\left(-1\right)^{n}\binom{N-1}{n}x^{n}}\,\mathrm{d}x}+\frac{N+1}{2N+1}\\ &=-N\int_{0}^{1}{x\left(1-x\right)^{2N-1}\,\mathrm{d}x}+\frac{N+1}{2N+1}\\ &=-\frac{1}{2\left(2N+1\right)}+\frac{N+1}{2N+1}\\ \sum_{n=0}^{N}{\frac{\left(-1\right)^{n}\left(N!\right)^{2}}{\left(N+n\right)!\left(N-n\right)!}}&=\frac{1}{2}\end{aligned}

Hence, $$ \sum_{n=0}^{N}\frac{(-1)^n}{{(N+n)!}(N-n)!}=\frac{1}{2\left(N!\right)^2} $$

Answer 2

Here is a combinatorial proof of $$\sum_{n=0}^N\,\frac{(-1)^n}{(N+n)!\,(N-n)!}=\frac{1}{2\,(N!)^2}\,.$$ Let $\mathcal{C}$ be the collection of subsets of $S:=\{1,2,\ldots,2N\}$ of size $N$, and $\mathcal{F}\subseteq \mathcal{C}$ be the subcollection containing subsets that do not have $2N$ as an element. It is easy to show that $$|\mathcal{F}|=\frac12\,\binom{2N}{N}\,.$$ (Each $T\in\mathcal{C}$ can be paired with its complement $S\setminus T$. Only one set of each complementary pair contains $2N$.)

For any subset $T$ of $S$, we define $$T^*:=\left\{\begin{array}{cc} T\cup\{2N\}&\text{if }2N\notin T\,,\\ T\setminus\{2N\}&\text{if }2N\in T\,. \end{array}\right.$$ For $n=0,1,2,\ldots,N$, let $\mathcal{T}_n$ be the collection of subsets of $S$ of size $N-n$. If $n$ is an integer greater than $N$, then set $\mathcal{T}_n:=\emptyset$. Clearly, $$\big|\mathcal{T}_n\big|=\binom{2N}{N-n}$$ for all $n=0,1,2,\ldots$.

Define $$\mathcal{T}_n^*:=\big\{T^*\,\big|\,T\in \mathcal{T}_n\big\}$$ for $n=0,1,2,\ldots$ (we actually need only the definitions of $\mathcal{T}_1^*$, $\mathcal{T}_3^*$, $\mathcal{T}_5^*$, $\ldots$ in this proof). We also set $$\mathcal{T}_n^-:=\big\{T\in\mathcal{T}_n\,\big|\,2N\notin T\big\}$$ and $$\mathcal{T}_n^+:=\big\{T\in\mathcal{T}_n\,\big|\,2N\in T\big\}$$ for $n=0,1,2,\ldots$. (This proof only requires the definitions of $\mathcal{T}^{\pm}_0$, $\mathcal{T}^{\pm}_2$, $\mathcal{T}^{\pm}_4$, $\ldots$.)

Observe that, for every positive integer $n$, $$\mathcal{T}_n^*=\mathcal{T}_{n-1}^+\sqcup \mathcal{T}^-_{n+1}\,.$$
Note that $\mathcal{F}$ is precisely $\mathcal{T}_0^-$. Ergo, $$\begin{align}\sum_{n=0}^1\,(-1)^n\,\big|\mathcal{T}_n\big| &=\big|\mathcal{T}_0\big|-\big|\mathcal{T}_\big|=\big|\mathcal{T}_0\big|-\big|\mathcal{T}_1^*\big| \\&=\big|\mathcal{T}_0\big|-\Big(\big|\mathcal{T}^+_{0}\big|+\big|\mathcal{T}^-_{2}\big|\Big) \\&=\Big(\big|\mathcal{T}_0\big|-\big|\mathcal{T}_0^+\big|\Big)-\big|\mathcal{T}^-_2\big| \\&=\big|\mathcal{T}^-_0\big|-\big|\mathcal{T}^-_2\big|=|\mathcal{F}|-\big|\mathcal{T}^-_2\big|\,.\end{align}$$ Therefore, $$\begin{align}\sum_{n=0}^2\,(-1)^n\,\big|\mathcal{T}_n\big| &=\sum_{n=0}^1\,(-1)^n\,\big|\mathcal{T}_n\big|+\big|\mathcal{T}_2\big| \\&=\Big(|\mathcal{F}|-\big|\mathcal{T}^-_2\big|\Big)+\big|\mathcal{T}_2\big| \\&=|\mathcal{F}|+\Big(|\mathcal{T}_2\big|-\big|\mathcal{T}^-_2\big|\Big)=|\mathcal{F}|+\big|\mathcal{T}_2^+\big|\,.\end{align}$$

In general, for $k=0,1,2,\ldots$, $$\sum_{n=0}^{2k-1}\,(-1)^n\,\big|\mathcal{T}_n\big|=|\mathcal{F}|-\big|\mathcal{T}_{2k}^-\big|\,,$$ and $$\sum_{n=0}^{2k}\,(-1)^n\,\big|\mathcal{T}_n\big|=|\mathcal{F}|+\big|\mathcal{T}_{2k}^+\big|\,.$$ This can be easily proven by induction on $k$.

By taking $k$ to be sufficiently large (say $k> \dfrac{N}{2}$), $\mathcal{T}_{2k}=\emptyset$ so that $$\frac{1}{2}\,\binom{2N}{N}=|\mathcal{F}|=|\mathcal{F}|\pm\big|\mathcal{T}_{2k}^\pm\big|=\sum_{n=0}^{N}\,(-1)^n\,\big|\mathcal{T}_n\big|=\sum_{n=0}^N\,(-1)^n\,\binom{2N}{N-n}\,.$$ This is equivalent to the desired equality.

Answer 3

Multiplying both sides by $2(2N)!$, we see the equality is equivalent to

\begin{align} &2\sum_{n=0}^{N}(-1)^n\binom{2N}{{N+n}}=\binom{2N}{N}. \end{align} Now write the left side \begin{align} 2\sum_{n=0}^{N}(-1)^n\binom{2N}{{N+n}} &=\left(\binom{2N}{N}-\binom{2N}{N+1}+\dots\right) +\left(\binom{2N}{N}-\binom{2N}{N+1}+\dots\right)\\ &=\left(\binom{2N}{N}-\binom{2N}{N-1}+\dots\right)+\left(\binom{2N}{N}-\binom{2N}{N+1}+\dots\right)\\ &=(-1)^{N}\cdot\left(\binom{2N}{0}-\binom{2N}{1}+\dots+\binom{2N}{2N}\right) + \binom{2N}{N}\\ &=(-1)^{N}\cdot(1-1)^{2N} + \binom{2N}{N}\\ &=\binom{2N}{N}\\ \end{align}