# Thread: evaluating an infinite product

1. ## evaluating an infinite product

How do I evaluate $\displaystyle \prod_{i=1}^\infty \frac{10^i - 1}{10^i} = .9\cdot .99 \cdot .99 \cdot \ldots$? Note that it's equivalent to solving the recurrence relation $\displaystyle p_{n+1} = p_n - \frac{p_n}{10^{n+1}}$ for $\displaystyle p_1 = \frac{9}{10}$.

I know how to prove it's between .8 and 1, but I can't figure out an exact answer.

2. Originally Posted by rn443
How do I evaluate $\displaystyle \prod_{i=1}^\infty \frac{10^i - 1}{10^i} = .9\cdot .99 \cdot .99 \cdot \ldots$? Note that it's equivalent to solving the recurrence relation $\displaystyle p_{n+1} = p_n - \frac{p_n}{10^{n+1}}$ for $\displaystyle p_1 = \frac{9}{10}$.

I know how to prove it's between .8 and 1, but I can't figure out an exact answer.
I don't think there is a closed formula. There is a famous formula by Euler that goes as follow : $\displaystyle \prod_{n=1}^\infty (1-q^n) = \frac{1}{\sum_{n=0}^\infty p(n) q^n}$ where $\displaystyle p(n)$ is the number of partitions of $\displaystyle n$. But no explicit formula is usually given; that says it all.

3. Originally Posted by Laurent
I don't think there is a closed formula. There is a famous formula by Euler that goes as follow : $\displaystyle \prod_{n=1}^\infty (1-q^n) = \frac{1}{\sum_{n=0}^\infty p(n) q^n}$ where $\displaystyle p(n)$ is the number of partitions of $\displaystyle n$. But no explicit formula is usually given; that says it all.
There is some more information about this function here. That link gives the expansion of $\displaystyle \prod_{n=1}^\infty (1-x^n)$ as $\displaystyle 1 - x - x^2 + x^5 + x^7 - x^{12} - x^{15} + x^{22} + x^{26} - x^{35} - x^{40} + x^{51} + \ldots$. It's not stated in the link, but it looks very much as though the nonzero coefficients are all $\displaystyle \pm1$ (with a pattern of two –1s followed by two +1s). If so then $\displaystyle \prod_{n=1}^\infty (1-10^{-n}) = 0.8900101 - 10^{-12} - 10^{-15}$ with an error of less than $\displaystyle 10^{-21}$.

In fact, that pattern of pairs of ±1s looks extremely regular, with the k'th pair being the coefficients of $\displaystyle x^{p_k}$ and $\displaystyle x^{q_k}$, where $\displaystyle p_k=\tfrac12k(3k-1)$ and $\displaystyle q_k=\tfrac12k(3k+1)$. Again, that's just based on observations from the list of the first 200 coefficients in the Encyclopedia of Integer Sequences. I don't have any proof for it.

4. Originally Posted by Opalg
There is some more information about this function here. That link gives the expansion of $\displaystyle \prod_{n=1}^\infty (1-x^n)$ as $\displaystyle 1 - x - x^2 + x^5 + x^7 - x^{12} - x^{15} + x^{22} + x^{26} - x^{35} - x^{40} + x^{51} + \ldots$. It's not stated in the link, but it looks very much as though the nonzero coefficients are all $\displaystyle \pm1$ (with a pattern of two –1s followed by two +1s). If so then $\displaystyle \prod_{n=1}^\infty (1-10^{-n}) = 0.8900101 - 10^{-12} - 10^{-15}$ with an error of less than $\displaystyle 10^{-21}$.

In fact, that pattern of pairs of ±1s looks extremely regular, with the k'th pair being the coefficients of $\displaystyle x^{p_k}$ and $\displaystyle x^{q_k}$, where $\displaystyle p_k=\tfrac12k(3k-1)$ and $\displaystyle q_k=\tfrac12k(3k+1)$. Again, that's just based on observations from the list of the first 200 coefficients in the Encyclopedia of Integer Sequences. I don't have any proof for it.
You seem to be correct. It is called Euler's pentagonal theorem and two proofs are given here.

5. A reasonable alternative to the 'direct computation' of the 'infinite product' is the computation of its logarithm...

$\displaystyle L(x)= \ln \prod_{n=1}^{\infty} (1-x^{n}) = \sum_{n=1}^{\infty} \ln (1-x^{n})$ (1)

Using the standard Taylor expansion of $\displaystyle \ln (1-x^{n})$ we obtain...

$\displaystyle L(x)= -\sum_{n=1}^{\infty} \sum_{k=1}^{\infty} \frac{x^{kn}}{k} = \sum_{k=1}^{\infty}\frac {x^{k}}{k\cdot (x^{k}-1)}$ (2)

Both the 'infinite product' and the 'infinite sum' are convergent for $\displaystyle |x|<1$. The computation using the first three hundred terms of the 'product' and the first three hundred terms of the 'sum' leads us to the same result approximated in twelve digits...

$\displaystyle \prod_{n=1}^{\infty} (1-.1^{n}) \approx .890010099999...$

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$

6. Originally Posted by chisigma
A reasonable alternative to the 'direct computation' of the 'infinite product' is the computation of its logarithm...

$\displaystyle L(x)= \ln \prod_{n=1}^{\infty} (1-x^{n}) = \sum_{n=1}^{\infty} \ln (1-x^{n})$ (1)

Using the standard Taylor expansion of $\displaystyle \ln (1-x^{n})$ we obtain...

$\displaystyle L(x)= -\sum_{n=1}^{\infty} \sum_{k=1}^{\infty} \frac{x^{kn}}{k} = \sum_{k=1}^{\infty}\frac {x^{k}}{k\cdot (x^{k}-1)}$ (2)

Both the 'infinite product' and the 'infinite sum' are convergent for $\displaystyle |x|<1$. The computation using the first three hundred terms of the 'product' and the first three hundred terms of the 'sum' leads us to the same result approximated in twelve digits...

$\displaystyle \prod_{n=1}^{\infty} (1-.1^{n}) \approx .890010099999...$

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$
A few related question please -
1. Is there a easy way to show if the result is rational or irrational?
2. @rn443: Can you please share how you proved that the result is greater than 0.8?

Thanks

7. Originally Posted by aman_cc
A few related question please -
1. Is there a easy way to show if the result is rational or irrational?
Yes.

Because of the formula

$\displaystyle \prod_{n=1}^\infty (1+q^n)=1+\sum_{n=1}^\infty (-1)^n (q^{\frac{n(3n-1)}{2}}+q^{\frac{n(3n+1)}{2}})$,

we have $\displaystyle \alpha=\prod_{n=1}^\infty \left(1+\frac{1}{10^n}\right) =$ $\displaystyle 1.00001010000000000000010001\ldots - 0.11000000000100100\ldots$ (only ones and zeros).

Note as well that $\displaystyle 0.111111\cdots = \frac{1}{9}$. Therefore,

$\displaystyle \alpha + \frac{1}{9} = 1.0011212111101101111112111211\cdots$.

There may be mistakes, but you see the idea: $\displaystyle \alpha+\frac{1}{9}$ has a decimal expansion with only 0,1,2's. And because the exponents in the first formula are quadratic (not linear), this expansion is not eventually periodic.

We conclude that $\displaystyle \alpha+\frac{1}{9}$ is irrational, and thus so is $\displaystyle \alpha$.