# Solving the Limit of sin(\pi*n/4)*\Gamma(x) Problem

• SumThePrimes
In summary, the limit as n goes to 0 of sin(\pi*n/4)*\Gamma(x) is equal to \pi/4, and this can be proven using the properties of the gamma function and L'hopital's rule.
SumThePrimes
I've recently been confronted with the limit as n goes to 0 of sin($\pi$*n/4)*$\Gamma$(x) , and have no idea on how to confront the problem, as I have little familiarity with the gamma function. Is there any relatively easy ways to prove this, or at least ways that use methods not difficult to learn? I would very much like to see a proof, as wolfram alpha gives a answer of $\pi$/4, and the answer is important relating to some very interesting alternating series.
edit 2: Wow, I haven't solved this problem, but if wolfram alpha is right, soon I'll be summing alternating series never summed before :) Well, ones I've never seen summed before at least...

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##\lim_{n \to 0} \sin(\frac{\pi n}{4}) \Gamma(x) = \sin(0)\Gamma(x) = 0##...

I suspect there is a typo in the original question - it should be Γ(n), not Γ(x).

It's actually quite easy. You know that the gamma function satisfies the functional equation $\Gamma(x+1) = x \Gamma(x)$, so then:

\begin{align*} \lim_{x \rightarrow 0} \sin (\pi x/4) \Gamma(x) &= \lim_{x \rightarrow 0} \frac{\sin(\pi x/4) \Gamma(x+1)}{x} \\ &= \left( lim_{x \rightarrow 0} \frac{\sin(\pi x/4)}{x} \right) \left( \lim_{x \rightarrow 0} \Gamma(x+1) \right) \\ &= lim_{x \rightarrow 0} \frac{\sin(\pi x/4)}{x}\end{align*}

Where the last equality follows from the fact that $\Gamma$ is continuous and $\Gamma(1) = 1$. But then by L'hopital's rule:

$$\lim_{x \rightarrow 0} \frac{\sin(\pi x/4)}{x} = \lim_{x \rightarrow 0} \frac{\pi}{4} \cos (\pi x/4) = \frac{\pi}{4}$$

## 1. What is the "Solving the Limit of sin(π*n/4)*Γ(x) Problem"?

The "Solving the Limit of sin(π*n/4)*Γ(x) Problem" is a mathematical problem that involves finding the limit of the function sin(π*n/4)*Γ(x) as n approaches infinity.

## 2. What is the significance of this problem?

This problem is important in the field of mathematics because it is used to study the behavior of functions as their inputs approach infinity. It also has applications in physics and engineering.

## 3. How do you solve this problem?

There are several methods for solving this problem, including using the properties of limits, the squeeze theorem, and the definition of the Gamma function. It is also possible to use mathematical software or calculators to approximate the limit.

## 4. Can you provide an example of solving this problem?

Sure, let's say we want to find the limit of sin(π*n/4)*Γ(x) as n approaches infinity. Using the squeeze theorem, we can show that this limit is equal to 0, since the function is bounded between -1 and 1 and the Gamma function approaches infinity faster than n as n approaches infinity.

## 5. Are there any real-world applications of this problem?

Yes, this problem has applications in various fields, such as physics, engineering, and economics. For example, it can be used to model the growth of populations or the decay of radioactive materials.

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