# 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.
The text passage contains a mathematical problem and its solution, specifically a proof that a given sequence is not bounded above. The problem is stated in the first post, and a detailed solution is provided in the second post.
single_question_with_answer
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# 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.
user
The text passage contains a mathematical problem and its solution, specifically a proof that a given sequence is not bounded above. The problem is stated in the first post, and a detailed solution is provided in the second post.

TYPE: SINGLE_QUESTION_WITH_ANSWER