MHB Evaluating an Infinite Sum of Binomial Coefficients

AI Thread Summary
The discussion centers on evaluating the infinite sum of binomial coefficients, specifically $\sum_{n=2009}^{\infty} \frac{1}{\binom{n}{2009}}$. Participants confirm the correctness of a solution provided by Pranav and explore an alternative method involving a telescoping sum. A detailed mathematical derivation is shared, illustrating the relationship between binomial coefficients and the evaluation of the sum. The conversation emphasizes clarity in presenting solutions to avoid confusion among readers. Overall, the exchange highlights effective strategies for tackling complex binomial coefficient sums.
anemone
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Evaluate $\displaystyle\lower0.5ex{\mathop{\large \sum}_{n=2009}^{\infty}} \dfrac{1}{n \choose 2009}$.
 
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anemone said:
Evaluate $\displaystyle \sum_{n=2009}^{\infty} \dfrac{1}{n \choose 2009}$.

Rewrite the sum as:
$$\sum_{n=2009}^{\infty} \frac{1}{n\choose 2009}=(2009)!\sum_{n=2009}^{\infty} \frac{(n-2009)!}{n!}=(2009)!\sum_{n=2009}^{\infty} \frac{1}{n(n-1)(n-2)(n-3)\cdots (n-2008)}$$
$$=(2009)!\sum_{n=0}^{\infty} \frac{1}{(n+1)(n+2)(n+3)\cdots (n+2009)}$$
To evaluate the above sum, we have to integrate the following 2009 times:
$$\frac{1}{1-x}=1+x+x^2+x^3+\cdots$$
I am unsure if the following is going to be correct. To integrate the above 2009 times, I used the technique of repeated integral as shown here: Repeated Integral -- from Wolfram MathWorld

So from above, I have to find the following:
$$\int_0^x \frac{(x-t)^{2008}}{(1-t)(2008)!}\,dt$$
at $x=1$, i.e
$$\int_0^1 \frac{(1-t)^{2007}}{(2008)!}=\frac{1}{(2008)!(2008)}$$
Hence,
$$(2009)!\sum_{n=0}^{\infty} \frac{1}{(n+1)(n+2)(n+3)\cdots (n+2009)}=(2009)!\frac{1}{(2008)!(2008)}=\boxed{ \dfrac {2009}{2008}}$$
I am not sure if what I have done is correct. :confused:
 
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Pranav said:
Rewrite the sum as:
$$\sum_{n=2009}^{\infty} \frac{1}{n\choose 2009}=(2009)!\sum_{n=2009}^{\infty} \frac{(n-2009)!}{n!}=(2009)!\sum_{n=2009}^{\infty} \frac{1}{n(n-1)(n-2)(n-3)\cdots (n-2008)}$$
$$=(2009)!\sum_{n=0}^{\infty} \frac{1}{(n+1)(n+2)(n+3)\cdots (n+2009)}$$
To evaluate the above sum, we have to integrate the following 2009 times:
$$\frac{1}{1-x}=1+x+x^2+x^3+\cdots$$
I am unsure if the following is going to be correct. To integrate the above 2009 times, I used the technique of repeated integral as shown here: Repeated Integral -- from Wolfram MathWorld

So from above, I have to find the following:
$$\int_0^x \frac{(x-t)^{2008}}{(1-t)(2008)!}\,dt$$
at $x=1$, i.e
$$\int_0^1 \frac{(1-t)^{2007}}{(2008)!}=\frac{1}{(2008)!(2008)}$$
Hence,
$$(2009)!\sum_{n=0}^{\infty} \frac{1}{(n+1)(n+2)(n+3)\cdots (n+2009)}=(2009)!\frac{1}{(2008)!(2008)}=\boxed{ \dfrac {2009}{2008}}$$
I am not sure if what I have done is correct. :confused:

Well done, Pranav! I think what you have done is correct and that led to the correct answer as well.:cool:

The alternative way to evaluate this sum is to recognize that this sum is a telescoping sum and I will show you and others another good solution that I have seen somewhere online.

Observe that

$\begin{align*}\dfrac{k+1}{k} \left(\dfrac{1}{n-1 \choose k}-\dfrac{1}{n \choose k} \right)&=\dfrac{k+1}{k} \dfrac{{n \choose k}-{n-1 \choose k}}{{n \choose k}{n-1 \choose k}}\\&=\dfrac{k+1}{k} \dfrac{n-1 \choose k-1}{{n \choose k}{n-1 \choose k}}\\&=\dfrac{k+1}{k} \dfrac{(n-1)!k!k!(n-k-1)!(n-k)!}{n!(n-1)!(k-1)!(n-k)!}\\&=\dfrac{k+1}{k} \dfrac{k\cdot k! (n-k-1)!}{n!}\\&=\dfrac{1}{n-1 \choose k}\end{align*}$

Now, we apply this with $k=2008$ and sum across all $n$ from 2009 to $\infty$, and get

$\displaystyle\lower0.5ex{\mathop{\large \sum}_{n=2009}^{\infty}} \dfrac{1}{n \choose 2009}=\dfrac{2009}{2008} \lower0.5ex{\mathop{\large \sum}_{n=2009}^{\infty}} \left( \dfrac{1}{n-1 \choose 2008}-\dfrac{1}{n \choose 2008} \right)$

Notice that all terms from teh sum on the RHS cancel, except for the initial term of $\dfrac{1}{2008 \choose 2008}$, which is equal to 1, so we can conclude that

$\displaystyle\lower0.5ex{\mathop{\large \sum}_{n=2009}^{\infty}} \dfrac{1}{n \choose 2009}=\dfrac{2009}{2008}$
 
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anemone said:
Observe that

$\begin{align*}\dfrac{k+1}{k} \left(\dfrac{1}{n-1 \choose k}-\dfrac{1}{n \choose k} \right)&=\dfrac{k+1}{k} \dfrac{{n \choose k}-{n-1 \choose k}}{{n \choose k}{n-1 \choose k}}\\&=\dfrac{k+1}{k} \dfrac{n-1 \choose k-1}{{n \choose k}{n-1 \choose k}}\\&=\dfrac{k+1}{k} \dfrac{(n-1)!k!k!(n-k-1)!(n-k)!}{n!(n-1)!(k-1)!(n-k)!}\\&=\dfrac{k+1}{k} \dfrac{k\cdot k! (n-k-1)!}{n!}\\&=\dfrac{1}{n-1 \choose k}\end{align*}$
That looks like a nice approach. Thanks for sharing! :)
 
Observe that

$\begin{align*}\dfrac{k+1}{k} \left(\dfrac{1}{n-1 \choose k}-\dfrac{1}{n \choose k} \right)&=\dfrac{k+1}{k} \dfrac{{n \choose k}-{n-1 \choose k}}{{n \choose k}{n-1 \choose k}}\\&=\dfrac{k+1}{k} \dfrac{n-1 \choose k-1}{{n \choose k}{n-1 \choose k}}\\&=\dfrac{k+1}{k} \dfrac{(n-1)!k!k!(n-k-1)!(n-k)!}{n!(n-1)!(k-1)!(n-k)!}\\&=\dfrac{k+1}{k} \dfrac{k\cdot k! (n-k-1)!}{n!}\\&=\dfrac{1}{n-1 \choose k}\end{align*}$

Now, we apply this with $k=2008$ and sum across all $n$ from 2009 to $\infty$, and get

$\displaystyle\lower0.5ex{\mathop{\large \sum}_{n=2009}^{\infty}} \dfrac{1}{n \choose 2009}=\dfrac{2009}{2008} \lower0.5ex{\mathop{\large \sum}_{n=2009}^{\infty}} \left( \dfrac{1}{n-1 \choose 2008}-\dfrac{1}{n \choose 2008} \right)$

Notice that all terms from teh sum on the RHS cancel, except for the initial term of $\dfrac{1}{2008 \choose 2008}$, which is equal to 1, so we can conclude that

$\displaystyle\lower0.5ex{\mathop{\large \sum}_{n=2009}^{\infty}} \dfrac{1}{n \choose 2009}=\dfrac{2009}{2008}$
it should be:
$\begin{align*}\dfrac{k+1}{k} \left(\dfrac{1}{n-1 \choose k}-\dfrac{1}{n \choose k} \right)&=\dfrac{k+1}{k} \dfrac{{n \choose k}-{n-1 \choose k}}{{n \choose k}{n-1 \choose k}}\\&=\dfrac{k+1}{k} \dfrac{n-1 \choose k-1}{{n \choose k}{n-1 \choose k}}\\&=\dfrac{k+1}{k} \dfrac{(n-1)!k!k!(n-k-1)!(n-k)!}{n!(n-1)!(k-1)!(n-k)!}\\&=\dfrac{k+1}{k} \dfrac{k\cdot k! (n-k-1)!}{n!}\\&=\dfrac{1}{n \choose k+1}\end{align*}$
 
Yes, Albert, you're right. I will leave my original post as is so that we won't confuse the readers and I apologize to the community for not bringing enough attention when I composed the solution.(Tmi):o
 
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