# Sum of Multichoose Reciprocals

I stumbled across an interesting formula involving the multichoose function. Let $$m,n$$ be natural numbers with $$m > 1$$, and suppose we want to find the sum of the reciprocals of the first $$n$$ multichoose coefficients. After experimenting a little, I found this surprisingly elegant formula:

$\sum_{r = 1}^{n} \left(\!\!{r\choose m}\!\!\right)^{-1} = \dfrac{m}{m-1} \left(1 - \left(\!\!{m\choose n}\!\!\right)^{-1} \right)$

Furthermore, if we add them to infinity, the formula simplifies to $\sum_{r = 1}^{\infty} \left(\!\!{r\choose m}\!\!\right)^{-1} = \dfrac{m}{m-1}$

So my question is: can you think of a nice combinatorial proof of this formula (without using induction)?

For those of you who don't know, $$\left(\!\!{n\choose m}\!\!\right)$$ denotes the number of different ways to pick a bunch of $$m$$ smarties with $$n$$ distinct choices of colour, where order is not important but repetition of colours is allowed.

It can also be defined this way: $\left(\!\!{n\choose m}\!\!\right) = \binom{n+m-1}{m}$

Note by Ariel Gershon
3 years, 8 months ago

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$\sum_{r = 1}^{\infty} \left(\!\!{r\choose m}\!\!\right)^{-1} = \dfrac{m}{m-1}$

$$\displaystyle \left(\!\!{r \choose m}\!\!\right) = \binom{r+m - 1}{m}$$

$$\displaystyle \sum_{r = 1}^{\infty} \dfrac{1}{\binom{r+m - 1}{m}} = \dfrac{(r - 1)!\times m!}{(m+r - 1)!}$$

$$= \dfrac{m!}{(r)(r+1).......(m + r - 1)}$$

$$\dfrac{1}{(r)(m + r - 1)} = \dfrac{1}{m - 1} \left ( \dfrac{1}{r} - \dfrac{1}{m + r - 1} \right)$$

$$\dfrac{m!}{(r)(r+1).......(m + r - 1)} = \dfrac{m!}{m - 1}\left( \dfrac{1}{r(r+1)....(m+r-2)} - \dfrac{1}{(r+1)....(m + r - 1)}\right)$$

$$\dfrac{m!}{m - 1} \times \dfrac{1}{(m - 1)!} = \dfrac{m}{m - 1}$$

- 3 years, 8 months ago

Yes, that's the algebraic proof using telescoping series, which is in essence a form of induction.

Is there any other way of interpreting this formula? For example, if we could write it as a taylor series expansion?

A combinatorial proof seems less likely to me, because there is no clear interpretation of $$\left( {r \choose m } \right)$$, though I might be wrong.

Staff - 3 years, 8 months ago

Perhaps you're right that there's no good combinatorial proof. However the interpretation of $$\left(\!\!{n \choose m}\!\!\right)$$ is the number of different multisets of size $$m$$ (where order is not important but repetition is allowed) which contain elements chosen from $$\{1,2,3, ... n\}$$.

We can always convert the formula into one that only involves binomial coefficients: $\sum_{k = m}^n \dfrac{1}{{k\choose m}} = \dfrac{m}{m-1} \left(1 - \dfrac{1}{\binom{n}{m-1}}\right)$

- 3 years, 8 months ago

Hi guys I wish to improve my mathematics skill , so can anyone help me with it , how do I proceed with this idea ?

- 3 years, 8 months ago

Check out the wikis (in the practice section), and work on more problems!

Staff - 3 years, 8 months ago