- #1

juantheron

- 247

- 1

Evaluate $\displaystyle \lim_{n\rightarrow \infty}\prod^{n}_{k=1}\left(1+\frac{1}{4k^2-1}\right)$

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In summary, a limit product evaluation, denoted by the symbol $\displaystyle \infty$, is a mathematical concept used to describe the behavior of a function as the input values approach infinity. It is important because it allows us to understand the behavior of functions at extreme values and can be used to solve problems in calculus and other areas of mathematics. To calculate a limit product evaluation, you need to identify the function and input value, then use algebraic techniques or graphing software to observe the pattern of values. Common misconceptions include thinking it represents the actual value of the function at infinity and that a function must have a limit product evaluation to exist. In real life, limit product evaluation can be used to model and predict the behavior of systems

- #1

juantheron

- 247

- 1

Evaluate $\displaystyle \lim_{n\rightarrow \infty}\prod^{n}_{k=1}\left(1+\frac{1}{4k^2-1}\right)$

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- #2

MarkFL

Gold Member

MHB

- 13,288

- 12

\(\displaystyle L=\prod_{k=1}^{\infty}\left(\frac{4k^2}{4k^2-1}\right)\)

Euler's infinite product for the sine function states:

\(\displaystyle \frac{\sin(x)}{x}=\prod_{k=1}^{\infty}\left(1-\left(\frac{x}{k\pi}\right)^2\right)\)

Let \(\displaystyle x=\frac{\pi}{2}\):

\(\displaystyle \frac{\sin\left(\dfrac{\pi}{2}\right)}{\dfrac{\pi}{2}}=\prod_{k=1}^{\infty}\left(1-\left(\frac{\dfrac{\pi}{2}}{k\pi}\right)^2\right)\)

\(\displaystyle \frac{2}{\pi}=\prod_{k=1}^{\infty}\left(1-\left(\frac{1}{2k}\right)^2\right)=\prod_{k=1}^{\infty}\left(\frac{4k^2-1}{4k^2}\right)\)

\(\displaystyle \frac{\pi}{2}=\prod_{k=1}^{\infty}\left(\frac{4k^2}{4k^2-1}\right)\)

Hence:

\(\displaystyle L=\frac{\pi}{2}\)

This is known as Wallis' product. :D

- #3

juantheron

- 247

- 1

Thanks Markfl for Nice solution. I have solved it using Wall,s formula and sandwitch Theorem

A limit product evaluation, denoted by the symbol $\displaystyle \infty$, is a mathematical concept used to describe the behavior of a function as the input values approach infinity. It represents the maximum or minimum value that a function can approach as the input values get larger and larger.

Limit product evaluation is important because it allows us to understand the behavior of functions at extreme values. It can be used to determine the end behavior of a function, identify asymptotes, and solve problems in calculus and other areas of mathematics.

To calculate a limit product evaluation, you need to first identify the function and the input value that is approaching infinity. Then, you can use algebraic techniques or graphing software to evaluate the function at increasingly large input values and observe the pattern of the values. If the values are approaching a specific number, that number is the limit product evaluation.

One common misconception about limit product evaluation is that it represents the actual value of the function at infinity. In reality, it only describes the behavior of the function as the input values approach infinity. Another misconception is that a function must have a limit product evaluation in order to exist, but this is not always the case.

Limit product evaluation can be used in real life situations to model and predict the behavior of systems that involve continuously increasing or decreasing quantities. For example, it can be used to determine the maximum speed of a car in a drag race or the maximum height a ball can reach when thrown into the air.

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