text: # Thread: Prove that the sequence is not bounded above
  
  1. ## Prove that the sequence is not bounded above
  
  Prove that $a_n=\frac{1\cdot 3\cdots(2n+1)}{2\cdot 4\cdots(2n)}$ is not bounded above.
  
  2. ## Re: Prove that the sequence is not bounded above
  
  We use $\ln(1+x)\geq x-\frac{x^2}2$ for $x\geq 0$ to get $\ln a_n=\sum_{j=1}^n\ln\left(1+\frac 1{2j}\right)\geq \sum_{j=1}^n\frac 1{2j}-\frac 12\sum_{j=1}^n\frac 1{(2j)^2}=\frac 12\sum_{j=1}^n\frac 1j-\frac 18\sum_{j=1}^n\frac 1{j^2}$.
  Since $\lim_{n\to \infty}\sum_{j=1}^n\frac 1j=+\infty$ and $\lim_{n\to \infty}\sum_{j=1}^n\frac 1{j^2}<+\infty$, we get the result.
explanation: The document contains a math question and a detailed answer to that question. The question asks to prove that the sequence \( a_n = \frac{1 \cdot 3 \cdots (2n+1)}{2 \cdot 4 \cdots (2n)} \) is not bounded above. The answer provides a proof using logarithmic inequalities and properties of series.
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: # Thread: Prove that the sequence is not bounded above
  
  1. ## Prove that the sequence is not bounded above
  
  Prove that $a_n=\frac{1\cdot 3\cdots(2n+1)}{2\cdot 4\cdots(2n)}$ is not bounded above.
  
  2. ## Re: Prove that the sequence is not bounded above
  
  We use $\ln(1+x)\geq x-\frac{x^2}2$ for $x\geq 0$ to get $\ln a_n=\sum_{j=1}^n\ln\left(1+\frac 1{2j}\right)\geq \sum_{j=1}^n\frac 1{2j}-\frac 12\sum_{j=1}^n\frac 1{(2j)^2}=\frac 12\sum_{j=1}^n\frac 1j-\frac 18\sum_{j=1}^n\frac 1{j^2}$.
  Since $\lim_{n\to \infty}\sum_{j=1}^n\frac 1j=+\infty$ and $\lim_{n\to \infty}\sum_{j=1}^n\frac 1{j^2}<+\infty$, we get the result.
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document contains a math question and a detailed answer to that question. The question asks to prove that the sequence \( a_n = \frac{1 \cdot 3 \cdots (2n+1)}{2 \cdot 4 \cdots (2n)} \) is not bounded above. The answer provides a proof using logarithmic inequalities and properties of series.
  
  Label: question-and-answer