text: 1. ## period
  
  Hey,
  
  I'm trying to find the period of:
  
  $\displaystyle f(x)=(\cos(\pi\gamma))^{n} \hspace{0,5cm}, \gamma \in \mathbb{R}, n \in \mathbb{N}$
  
  It seems that when $\displaystyle n$ is even then the $\displaystyle f$ has period=1 and when $\displaystyle n$ is odd then $\displaystyle f$ has period=2 .
  
  How to show that analytically?
  
  2. As written, your function is not periodic. It's constant. Care to have another go?
  
  3. ## period
  
  Hey,
  
  My mistake. It should say:
  
  $\displaystyle f(x)=(\cos(\pi x))^{n} \hspace{0,5cm}, x \in \mathbb{R}, n \in \mathbb{N}$
  
  4. Must you consider more than this?
  
  $\displaystyle [cos(\pi x)]^{n+1} = [cos(\pi x)]^{n}\cdot cos(\pi x)$
  
  If you can make conclusions about even values of n, there is not much left for odd values.
  
  5. ## periodicity
  
  Hey,
  
  Sorry for the inconvenience. What I wanted do to was to arrive at the conclusion analytically rather than explanatory. How do I do that?
  
  6. What's the definition of "Period"?
  
  Something like: The minimum value of 'h' such that f(x+h) = f(x).
  
  Definitions are always good places to start. They are not always the best places to start writing programs, but for proving results, they are gold.
explanation: The document contains a series of posts discussing a math problem related to finding the period of a function. The user initially presents a function and asks how to show the period analytically. There are multiple responses, including corrections and suggestions for approaching the problem. The document does not contain a complete question and answer but rather a discussion that builds towards a solution.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: 1. ## period
  
  Hey,
  
  I'm trying to find the period of:
  
  $\displaystyle f(x)=(\cos(\pi\gamma))^{n} \hspace{0,5cm}, \gamma \in \mathbb{R}, n \in \mathbb{N}$
  
  It seems that when $\displaystyle n$ is even then the $\displaystyle f$ has period=1 and when $\displaystyle n$ is odd then $\displaystyle f$ has period=2 .
  
  How to show that analytically?
  
  2. As written, your function is not periodic. It's constant. Care to have another go?
  
  3. ## period
  
  Hey,
  
  My mistake. It should say:
  
  $\displaystyle f(x)=(\cos(\pi x))^{n} \hspace{0,5cm}, x \in \mathbb{R}, n \in \mathbb{N}$
  
  4. Must you consider more than this?
  
  $\displaystyle [cos(\pi x)]^{n+1} = [cos(\pi x)]^{n}\cdot cos(\pi x)$
  
  If you can make conclusions about even values of n, there is not much left for odd values.
  
  5. ## periodicity
  
  Hey,
  
  Sorry for the inconvenience. What I wanted do to was to arrive at the conclusion analytically rather than explanatory. How do I do that?
  
  6. What's the definition of "Period"?
  
  Something like: The minimum value of 'h' such that f(x+h) = f(x).
  
  Definitions are always good places to start. They are not always the best places to start writing programs, but for proving results, they are gold.
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document contains a series of posts discussing a math problem related to finding the period of a function. The user initially presents a function and asks how to show the period analytically. There are multiple responses, including corrections and suggestions for approaching the problem. The document does not contain a complete question and answer but rather a discussion that builds towards a solution.
  
  Label: multiple-question-and-answer