Picking $k_n$: Understanding the Corollary

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In summary, the conversation discusses the selection of a value for $k_n$ in the equation $\sin\pi z = \prod\limits_{n\in\mathbb{Z}-\{0\}}\left[\left(1-\frac{z}{n}\right)e^{z/n}\right]$ and the relationship between $k_n$ and the convergence of the infinite product. The equation is compared to the infinite product representation of the Gamma function and the corollary that $\sum\limits_{n =1}^{\infty}\frac{1}{\left|n\right|^2}$ converges. Some confusion is expressed about the meaning and limits of the product and the correct representation of $\sin
  • #1
Dustinsfl
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I don't understand how one picks a $k_n$.

For example, let's look at $\sin\pi z = \prod\limits_{n\in\mathbb{Z}-\{0\}}\left[\left(1-\frac{z}{n}\right)e^{z/n}\right]$.

For all n, $k_n = 2$. With this $k_n$, the product is entire. What is $k_n$? I know for that product we can write it as $\prod\limits_{n=1}^{\infty}\left(1-\frac{z^2}{n^2}\right)$. So $k_n$ is the power of z and n. I don't know why this was chosen.

Then there is the corollary:
If $\sum\limits_{n =1}^{\infty}\frac{1}{\left|n\right|^2}$ converges, then $\prod\limits_{n=1}^{\infty}\left[\left(1-\frac{z}{n}\right)e^{z/n}\right]$ converges.

I guessing this all related to picking $k_n$ but I don't get it.
 
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  • #2
dwsmith said:
I don't understand how one picks a $k_n$.

For example, let's look at $\sin\pi z = \prod\limits_{n\in\mathbb{Z}-\{0\}}\left[\left(1-\frac{z}{n}\right)e^{z/n}\right]$.

For all n, $k_n = 2$. With this $k_n$, the product is entire. What is $k_n$? I know for that product we can write it as $\prod\limits_{n=1}^{\infty}\left(1-\frac{z^2}{n^2}\right)$. So $k_n$ is the power of z and n. I don't know why this was chosen.

Then there is the corollary:
If $\sum\limits_{n =1}^{\infty}\frac{1}{\left|n\right|^2}$ converges, then $\prod\limits_{n=1}^{\infty}\left[\left(1-\frac{z}{n}\right)e^{z/n}\right]$ converges.

I guessing this all related to picking $k_n$ but I don't get it.

Hi dwsmith,

I don't understand how one picks a $k_n$.

What is \(k_n\)? Can you please give us more information.

For example, let's look at $\sin\pi z = \prod\limits_{n\in\mathbb{Z}-\{0\}}\left[\left(1-\frac{z}{n}\right)e^{z/n}\right]$.

How did you get this equation, and what are the limits of the infinite product? The Gamma function has a infinite product representation similar to this, but I have never seen a representation of this kind for \(\sin\pi z\).

Then there is the corollary:
If $\sum\limits_{n =1}^{\infty}\frac{1}{\left|n\right|^2}$ converges, then $\prod\limits_{n=1}^{\infty}\left[\left(1-\frac{z}{n}\right)e^{z/n}\right]$ converges.

This has no meaning since \(\sum\limits_{n =1}^{\infty}\frac{1}{\left|n\right|^2}=\sum\limits_{n =1}^{\infty}\frac{1}{n^2}\) is always convergent.

I know for that product we can write it as \(\prod\limits_{n=1}^{\infty}\left(1-\frac{z^2}{n^2}\right)\,.\)

\(\displaystyle\sin \pi z = \pi z \prod_{n=1}^{\infty} \left(1 - \frac{z^2}{n^2}\right)\) not \(\displaystyle\prod_{n=1}^{\infty} \left(1 - \frac{z^2}{n^2}\right)\).
 

FAQ: Picking $k_n$: Understanding the Corollary

What is the purpose of "Picking $k_n$?"

The purpose of "Picking $k_n$" is to understand the relationship between the sample size and the sample mean. It helps us determine the appropriate sample size to use in order to accurately estimate a population parameter.

What is the Corollary in "Picking $k_n$?"

The Corollary in "Picking $k_n$" states that as the sample size increases, the variability of the sample mean also decreases. In other words, the larger the sample size, the more accurate our estimate of the population mean will be.

How do I determine the minimum sample size needed for a desired level of precision?

The minimum sample size needed can be determined by using the formula $n = \frac{(z_{\alpha/2}\sigma)^2}{E^2}$, where $n$ is the sample size, $z_{\alpha/2}$ is the critical value for the desired confidence level, $\sigma$ is the population standard deviation, and $E$ is the desired margin of error.

Is "Picking $k_n$" applicable to all types of data?

Yes, "Picking $k_n$" is applicable to all types of data as long as the data follows a normal distribution. If the data does not follow a normal distribution, other methods, such as the central limit theorem, can be used to estimate the population mean.

Why is it important to understand "Picking $k_n$?"

Understanding "Picking $k_n$" is important because it allows us to make accurate inferences about a population based on a sample. It helps us determine the appropriate sample size to use in order to reduce sampling error and increase the precision of our estimates.

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