Binomial Coefficient Challenge 6!

Prove the following -

$\displaystyle \begin{equation*} 1 - \cfrac{1}{n+\cfrac{2n}{n-3+\cfrac{3(n-1)}{n-5+\cfrac{4(n-2)}{\ddots \cfrac{(m-1)(n-m+3)}{n-(2m-3)}}}}} \end{equation*} = \sum_{0\le k <m}{\dfrac{{(-1)}^k}{\binom{n}{k}}} = \dfrac{n+1}{n+2}\left(1 + \frac{{(-1)}^{m+1}}{\binom{n+1}{m}}\right)$

The continued fraction part is optional. It is just a bonus.

This is completely original.

Easy Math Editor

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Comments

$\text{S}=\displaystyle\sum_{r=0}^{m-1}\dfrac{(-1)^r}{\dbinom{n}{r}}$

$=\displaystyle\sum_{r=0}^{m-1}(-1)^r\dfrac{r!(n-r)!}{n!}$

$=\displaystyle\sum_{r=0}^{m-1}(-1)^r \dfrac{r!(n-r)!\color{#3D99F6}{(n-r+1+r+1)}}{n!}\color{#D61F06}{\times\dfrac{1}{(n+2)}}$

$=\displaystyle\sum_{r=0}^{m-1}\dfrac{1}{n!(n+2)}\left[(-1)^r\left\{r!(n-r+1)!+(n-r)!(r+1)!\right\}\right]$

$=\displaystyle\sum_{r=0}^{m-1}\dfrac{1}{n!(n+2)}\left[T_{r} - T_{r+1}\right]$

where $T_{r} = (-1)^r r!(n-r+1)!$

Evaluating using telescoping the sum, we have,

$\text{S} = \dfrac{n+1}{n+2}\left(1+\dfrac{(-1)^{m+1}}{\dbinom{n+1}{m}}\right)$

Some other methods can be to use Beta functions or using the result (from Partial Fractions)

$\dfrac{1}{\dbinom{n}{k}}=\sum_{j=1}^k(-1)^{k-j}\binom{k}{j}\dfrac{j}{n-j+1}$

The continued fraction result is fascinating! I'll work on it when I get some time.

Ishan Singh - 4 years ago

@Kartik Sharma Btw, what was your approach to this problem?

Ishan Singh - 4 years ago

I used Finite Calculus(Gosper's Method). I thought of beta functions but it was tedious. Telescoping serious was very nice although a bit too tricky.

I will add my method later. For the continued fraction part if you are asking, then I used a standard result in the theory of evaluating continued fractions.

Kartik Sharma - 4 years ago

@Kartik Sharma – I was talking about the summation. I assume this Gosper's method?

I have added some explanation in the telescoping part for clarity

Ishan Singh - 4 years ago

@Ishan Singh – Yes. It is quite general.

Kartik Sharma - 4 years ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$