How Is the Infinite Product of a Recurrence Sequence Solved?

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  • Thread starter Ackbach
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    2017
In summary, the POTW stands for "Problem of the Week" and is a weekly challenge or question that requires critical thinking skills to solve. The solution is chosen based on accuracy, logic, and evidence, and there can be multiple correct solutions. To improve chances of solving the POTW, it is recommended to practice and collaborate with others, and submissions are evaluated based on the same criteria.
  • #1
Ackbach
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Here is this week's POTW:

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Let $a_0 = \dfrac52$ and $a_k = a_{k-1}^2 - 2$ for $k \geq 1$. Compute $\displaystyle\prod_{k=0}^\infty \left(1 - \frac{1}{a_k} \right)$ in closed form.

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  • #2
No one answered this week's POTW, which was Problem A-3 in the 2014 Putnam Archives. The solution, attributed to Kiran Kedlaya and associates, follows.

[sp]
Using the identity
\[
(x + x^{-1})^2 - 2 = x^2 + x^{-2},
\]
we may check by induction on $k$ that $a_k = 2^{2^k} + 2^{-2^k}$; in particular, the product is absolutely convergent. Using the identities
\begin{align*}
\frac{x^2 + 1 + x^{-2}}{x + 1 + x^{-1}} &= x - 1 + x^{-1}, \\
\frac{x^2 - x^{-2}}{x - x^{-1}} &= x + x^{-1},
\end{align*}
we may telescope the product to obtain
\begin{align*}
\prod_{k=0}^\infty \left( 1 - \frac{1}{a_k} \right)
&= \prod_{k=0}^\infty \frac{2^{2^k} - 1 + 2^{-2^k}}{2^{2^k} + 2^{-2^k}} \\
&= \prod_{k=0}^\infty \frac{2^{2^{k+1}} + 1 + 2^{-2^{k+1}}}{2^{2^k} + 1 + 2^{-2^k}} \cdot
\frac{2^{2^k} - 2^{-2^k}}{2^{2^{k+1}} - 2^{2^{-k-1}}} \\
&= \frac{2^{2^0} - 2^{-2^0}}{2^{2^0}+1 + 2^{-2^0}} = \frac{3}{7}.
\end{align*}
[/sp]
 

FAQ: How Is the Infinite Product of a Recurrence Sequence Solved?

What is the POTW and what does it stand for?

The POTW stands for "Problem of the Week" and it is a weekly challenge or question that is posed to individuals or groups to solve using their knowledge and critical thinking skills.

How is the solution to the POTW chosen?

The solution to the POTW is chosen based on the most accurate and logical answer that is supported by evidence and reasoning. In some cases, there may be multiple correct solutions, but the most commonly accepted solution is chosen.

Is there only one correct solution to the POTW?

No, there can be multiple correct solutions to the POTW. However, the most commonly accepted solution is usually chosen as the official answer.

How can I improve my chances of solving the POTW?

The best way to improve your chances of solving the POTW is to constantly practice and build your knowledge and critical thinking skills. It is also helpful to collaborate with others and discuss different approaches to solving the problem.

Can I submit my own solution to the POTW?

Yes, you can submit your own solution to the POTW. However, it will be evaluated based on its accuracy, logic, and supporting evidence, just like any other submission.

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