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# binomial formula for negative integer powers

For negative integer powers, the binomial formula can be written in terms of binomial coefficients like so:

$(1-x)^{{-n}}=\sum_{{m=1}}^{\infty}\binom{m+n-1}{n-1}x^{m}$ |

Proof: We shall prove this by induction on $n$. First, note that, if $n=1$, then $\binom{m}{0}=1$, so our formula reduces to

$(1-x)^{{-1}}=\sum_{{m=1}}^{\infty}x^{m},$ |

which is the formula for the sum of an infinite geometric series.

Next, suppose that the formula is valid for a certain value of $n$. Then we have

$(1-x)^{{-n-1}}=(1-x)^{{-1}}(1-x)^{{-n}}=\left(\sum_{{k=0}}^{\infty}x^{k}\right% )\left(\sum_{{m=0}}^{\infty}{m+n-1\choose n-1}x^{m}\right)$ |

The product of sums can be rewritten as the following double sum:

$\sum_{{m=0}}^{\infty}\sum_{{k=0}}^{m}{n+k-1\choose n-1}x^{m}$ |

The easiest way to see this is by rearranging the double sum as follows and adding columns

$\begin{matrix}x^{0}\sum_{{m=0}}^{\infty}\binom{m+n-1}{n-1}x^{m}=&\binom{n-1}{n% -1}&+&\binom{n}{n-1}x&+&\binom{n+1}{n-1}x^{2}&+&\binom{n+2}{n-1}x^{3}&+&\binom% {n+3}{n-1}x^{4}&+&\cdots\\ x^{1}\sum_{{m=0}}^{\infty}\binom{m+n-1}{n-1}x^{m}=&&&\binom{n-1}{n-1}x&+&% \binom{n}{n-1}x^{2}&+&\binom{n+1}{n-1}x^{3}&+&\binom{n+2}{n-1}x^{4}&+&\cdots\\ x^{2}\sum_{{m=0}}^{\infty}\binom{m+n-1}{n-1}x^{m}=&&&&&\binom{n-1}{n-1}x^{2}&+% &\binom{n}{n-1}x^{3}&+&\binom{n+1}{n-1}x^{4}&+&\cdots\\ x^{3}\sum_{{m=0}}^{\infty}\binom{m+n-1}{n-1}x^{m}=&&&&&&&\binom{n-1}{n-1}x^{3}% &+&\binom{n}{n-1}x^{4}&+&\cdots\\ .&.&.&.&.&.&.&.&.&.&.&.\end{matrix}$ |

To evaluate the finite sums, we shall use the following identity for binomial coefficients. (See the entry “binomial coefficient” for more information about this identity.)

$\sum_{{k=0}}^{m}\binom{n+k-1}{n-1}=\binom{m+n}{n}$ |

Inserting this result value for the finite sum back into the double sum, we obtain

$(1-x)^{{-n-1}}=\sum_{{m=0}}^{\infty}\binom{m+n}{n}x^{m}.$ |

Q.E.D.

## Mathematics Subject Classification

26A06*no label found*

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## Comments

## What is wrong here?

Could someone explain to me why, even though the big fat matrix in this entry is good TeX --- I ran it through TeX and it compiled just fine --- it won't show up in Planet Math either as HTML or as page image, but instead I get an error message about extra alignment tabs. (which doesn't make much sense to be because good old TeX seems to think it has the right number of alignment tabs.) Is it because I have binomial coefficients in the entries of the matrix or a bug in the program or what?

## Re: What is wrong here?

You are using the standard PlanetMath preamble, which among other stuff includes amsmath packages. The amsmath package will bitterly complain to using obsolete constructions like \choose. There are two solutions: first is to use \binom, and the other is to not include amsmath. I would recommend to use amsmath's tools because they are more structural and better supported by LaTeX2HTML, which is used to generate HTML version of the PlanetMath entries.

Boris

## Re: What is wrong here?

That couldn't be the problem, or at least not the whole problem, because I changed all the \choose's which occurred inside the matrix to \binom's and got the same error messare about extra alignment tabs.

## Re: What is wrong here?

Yes, you are right, \choose was not the entire problem. The reason seems to be the counter \MaxMatrixCols which specifies the maximal number of columns in a matrix. According to amsldoc (which is in doc/latex/amsmath directory of texmf tree) the default value of that counter is 10, and you have 12 columns in your matrix. So, add \addtocounter{MaxMatrixCols}{2} to the preamble, and it should work.

Boris

## Re: What is wrong here?

Thank you very much for taking the time to figure out what the problem was and explaining how to fix it. The entry now renders properly and I have learned a few little facts about AMS - LaTeX.