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Pythagoras0 (토론 | 기여) (새 문서: ==예== 월리스 곱 (Wallis product formula)의 증명과정에 나오는 수열 다음과 같이 수열을 정의하자 :<math>a_n:=\int_0^{\pi}\sin^{n}\theta{d\theta}</math> ...) |
Pythagoras0 (토론 | 기여) (→예) |
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(같은 사용자의 중간 판 3개는 보이지 않습니다) | |||
3번째 줄: | 3번째 줄: | ||
다음과 같이 수열을 정의하자 :<math>a_n:=\int_0^{\pi}\sin^{n}\theta{d\theta}</math> | 다음과 같이 수열을 정의하자 :<math>a_n:=\int_0^{\pi}\sin^{n}\theta{d\theta}</math> | ||
− | + | <math>a_n</math>은 다음 점화식을 만족시킨다 :<math>a_0=\pi</math> :<math>a_1=2</math> :<math>a_{n}=\frac{n-1}{n}a_{n-2} \label{rec}</math> | |
− | 다음과 같은 극한을 계산하는 문제 | + | 다음과 같은 극한을 계산하는 문제 :<math>\lim_{n\to \infty } \, \frac{a_{2 n}}{a_{2 n+1}}=1 \label{lim}</math> |
− | + | <math>a_{n}</math>은 단조감소수열이므로, 다음 부등식이 성립한다 | |
− | + | :<math>1 \le \frac{a_{2n}}{a_{2n+1}} \le \frac{a_{2n-1}}{a_{2n+1}}=\frac{2n+1}{2n}</math> | |
우변에서는 \ref{rec}이 사용되었다. | 우변에서는 \ref{rec}이 사용되었다. | ||
− | 따라서 [[샌드위치 정리]]에 의해 | + | 따라서 [[샌드위치 정리]]에 의해 :<math>\lim_{n\to \infty } \, \frac{a_{2 n}}{a_{2 n+1}}=1</math> |
+ | |||
+ | == 노트 == | ||
+ | |||
+ | ===말뭉치=== | ||
+ | # The squeeze theorem is used in calculus and mathematical analysis.<ref name="ref_3ab4d0d7">[https://en.wikipedia.org/wiki/Squeeze_theorem Squeeze theorem]</ref> | ||
+ | # There are many things that you could try when you see an absolute value in a problem, and the squeeze theorem is one of them.<ref name="ref_920f0335">[https://brilliant.org/wiki/squeeze-theorem/ Brilliant Math & Science Wiki]</ref> | ||
+ | # How to use the squeeze theorem?<ref name="ref_2f83a5e1">[https://calcworkshop.com/limits/squeeze-theorem/ Squeeze Theorem How-To w/ 4 Step-by-Step Examples!]</ref> | ||
+ | # In fact, that’s the whole idea behind the squeeze theorem, also known as the pinching theorem or the sandwich theorem.<ref name="ref_2f83a5e1" /> | ||
+ | # But with the help of the squeeze theorem, we can now determine the limit of an oscillating function!<ref name="ref_2f83a5e1" /> | ||
+ | # In other words, the squeeze theorem is a proof that shows the value of a limit by smooshing a tricky function between two equal and known values.<ref name="ref_2f83a5e1" /> | ||
+ | # This exercise explores the squeeze or sandwich theorem.<ref name="ref_5b5d906c">[https://khanacademy.fandom.com/wiki/Squeeze_theorem Squeeze theorem]</ref> | ||
+ | # The user is expected use the function to determine the limit via the squeeze theorem and provide the correct answer.<ref name="ref_5b5d906c" /> | ||
+ | # The squeeze theorem is also sometimes called the sandwich theorem.<ref name="ref_5b5d906c" /> | ||
+ | # The squeeze theorem is another way to solve for tricky limits.<ref name="ref_1ea88952">[http://www.xaktly.com/SqueezeTheorem.html Squeeze theorem]</ref> | ||
+ | # Now, before we look at some concrete examples of how and when to do squeeze theorem, let's first review how to find and evaluate limits.<ref name="ref_dfa175e3">[https://www.studypug.com/calculus-help/squeeze-theorem How to use the squeeze theorem]</ref> | ||
+ | # We will now proceed to specifically look at the limit squeeze theorem (law 7 from the Limit of a Sequence page) and prove it's validity.<ref name="ref_32f6b49f">[http://mathonline.wikidot.com/the-squeeze-theorem-for-convergent-sequences The Squeeze Theorem for Convergent Sequences]</ref> | ||
+ | # This result is also known, in the UK in particular, as the sandwich theorem or the sandwich rule.<ref name="ref_a17d7b4e">[https://proofwiki.org/wiki/Squeeze_Theorem Squeeze Theorem]</ref> | ||
+ | # So, in order to use the squeeze theorem on a limit, we just have to find functions similar enough that all three functions squeeze together at a particular point like the image below.<ref name="ref_bf63929d">[https://fiveable.me/ap-calc/unit-1/determining-limits-using-squeeze-theorem/study-guide/0Ax6y3Qku88ex24KGwiG Determining Limits Using the Squeeze Theorem]</ref> | ||
+ | # The squeeze theorem espresses in precise mathematical terms a simple idea.<ref name="ref_e805dd6f">[http://www.intuitive-calculus.com/squeeze-theorem.html Intuition Behind the Squeeze Theorem and Applications]</ref> | ||
+ | # You start to see how we'll use the squeeze theorem?<ref name="ref_e805dd6f" /> | ||
+ | # Now, we're ready to use the squeeze theorem!<ref name="ref_e805dd6f" /> | ||
+ | # When is the squeeze theorem applied?<ref name="ref_53b00c14">[https://math.stackexchange.com/questions/3306633/when-is-the-squeezesandwich-theorem-used When Is the Squeeze(Sandwich) Theorem Used?]</ref> | ||
+ | # Are there other such functions to apply when using the squeeze theorem?<ref name="ref_53b00c14" /> | ||
+ | # Squeeze Theorem (or also known as the sandwich theorem) uses two functions to find the limit of the actual function we’re working on.<ref name="ref_6cfa5e7f">[https://www.storyofmathematics.com/squeeze-theorem Squeeze theorem – Definition, Proof, and Examples]</ref> | ||
+ | # In calculus, the squeeze theorem (known also as the pinching theorem, the sandwich theorem, the sandwich rule and sometimes the squeeze lemma) is a theorem regarding the limit of a function.<ref name="ref_56d95407">[https://en.wikibooks.org/wiki/Undergraduate_Mathematics/Squeeze_theorem Undergraduate Mathematics/Squeeze theorem]</ref> | ||
+ | # The squeeze theorem is a technical result that is very important in proofs in calculus and mathematical analysis.<ref name="ref_56d95407" /> | ||
+ | # The squeeze theorem is formally stated as follows.<ref name="ref_56d95407" /> | ||
+ | # Some people call it the sandwich theorem, but I like the term squeeze.<ref name="ref_17389b01">[https://study.com/academy/lesson/squeeze-theorem-definition-and-examples.html Squeeze Theorem: Definition and Examples - Math Class (Video)]</ref> | ||
+ | # The squeezing theorem is also called the sandwich theorem.<ref name="ref_fbfc6d73">[https://mathworld.wolfram.com/SqueezingTheorem.html Squeezing Theorem -- from Wolfram MathWorld]</ref> | ||
+ | # Can we apply Squeeze theorem for the following limits?<ref name="ref_ab5ec754">[http://www.people.ku.edu/~s890m022/Math127_F18/Limits-Squeeze_Thm..pdf Math 127 – calculus iii]</ref> | ||
+ | # If a two variable function satisfy the requirements, then we may apply squeeze theorem.<ref name="ref_ab5ec754" /> | ||
+ | # 1 2 USING THE SQUEEZE THEOREM AND INTERMEDIATE VALUE THEOREM Claim.<ref name="ref_3241ae0d">[https://www.math.wisc.edu/~csimpson6/teaching/2018_fall_221/ivt/ivt.pdf Using the squeeze theorem and intermediate]</ref> | ||
+ | # To do this, well use the Squeeze theorem by establishing upper and lower bounds on sin(x)/x in an interval around 0.<ref name="ref_d9b24eea">[http://ime.math.arizona.edu/g-teams/Profiles/JS/Calc/SqueezeTheorem.pdf The squeeze theorem: statement and example]</ref> | ||
+ | ===소스=== | ||
+ | <references /> | ||
+ | |||
+ | == 메타데이터 == | ||
+ | |||
+ | ===위키데이터=== | ||
+ | * ID : [https://www.wikidata.org/wiki/Q1065257 Q1065257] | ||
+ | ===Spacy 패턴 목록=== | ||
+ | * [{'LOWER': 'squeeze'}, {'LEMMA': 'theorem'}] | ||
+ | * [{'LOWER': 'pinching'}, {'LEMMA': 'theorem'}] | ||
+ | * [{'LOWER': 'sandwich'}, {'LEMMA': 'theorem'}] | ||
+ | * [{'LOWER': 'sandwich'}, {'LEMMA': 'rule'}] | ||
+ | * [{'LOWER': 'squeeze'}, {'LEMMA': 'lemma'}] |
2021년 2월 26일 (금) 01:29 기준 최신판
예
월리스 곱 (Wallis product formula)의 증명과정에 나오는 수열
다음과 같이 수열을 정의하자 \[a_n:=\int_0^{\pi}\sin^{n}\theta{d\theta}\] \(a_n\)은 다음 점화식을 만족시킨다 \[a_0=\pi\] \[a_1=2\] \[a_{n}=\frac{n-1}{n}a_{n-2} \label{rec}\]
다음과 같은 극한을 계산하는 문제 \[\lim_{n\to \infty } \, \frac{a_{2 n}}{a_{2 n+1}}=1 \label{lim}\]
\(a_{n}\)은 단조감소수열이므로, 다음 부등식이 성립한다
\[1 \le \frac{a_{2n}}{a_{2n+1}} \le \frac{a_{2n-1}}{a_{2n+1}}=\frac{2n+1}{2n}\]
우변에서는 \ref{rec}이 사용되었다.
따라서 샌드위치 정리에 의해 \[\lim_{n\to \infty } \, \frac{a_{2 n}}{a_{2 n+1}}=1\]
노트
말뭉치
- The squeeze theorem is used in calculus and mathematical analysis.[1]
- There are many things that you could try when you see an absolute value in a problem, and the squeeze theorem is one of them.[2]
- How to use the squeeze theorem?[3]
- In fact, that’s the whole idea behind the squeeze theorem, also known as the pinching theorem or the sandwich theorem.[3]
- But with the help of the squeeze theorem, we can now determine the limit of an oscillating function![3]
- In other words, the squeeze theorem is a proof that shows the value of a limit by smooshing a tricky function between two equal and known values.[3]
- This exercise explores the squeeze or sandwich theorem.[4]
- The user is expected use the function to determine the limit via the squeeze theorem and provide the correct answer.[4]
- The squeeze theorem is also sometimes called the sandwich theorem.[4]
- The squeeze theorem is another way to solve for tricky limits.[5]
- Now, before we look at some concrete examples of how and when to do squeeze theorem, let's first review how to find and evaluate limits.[6]
- We will now proceed to specifically look at the limit squeeze theorem (law 7 from the Limit of a Sequence page) and prove it's validity.[7]
- This result is also known, in the UK in particular, as the sandwich theorem or the sandwich rule.[8]
- So, in order to use the squeeze theorem on a limit, we just have to find functions similar enough that all three functions squeeze together at a particular point like the image below.[9]
- The squeeze theorem espresses in precise mathematical terms a simple idea.[10]
- You start to see how we'll use the squeeze theorem?[10]
- Now, we're ready to use the squeeze theorem![10]
- When is the squeeze theorem applied?[11]
- Are there other such functions to apply when using the squeeze theorem?[11]
- Squeeze Theorem (or also known as the sandwich theorem) uses two functions to find the limit of the actual function we’re working on.[12]
- In calculus, the squeeze theorem (known also as the pinching theorem, the sandwich theorem, the sandwich rule and sometimes the squeeze lemma) is a theorem regarding the limit of a function.[13]
- The squeeze theorem is a technical result that is very important in proofs in calculus and mathematical analysis.[13]
- The squeeze theorem is formally stated as follows.[13]
- Some people call it the sandwich theorem, but I like the term squeeze.[14]
- The squeezing theorem is also called the sandwich theorem.[15]
- Can we apply Squeeze theorem for the following limits?[16]
- If a two variable function satisfy the requirements, then we may apply squeeze theorem.[16]
- 1 2 USING THE SQUEEZE THEOREM AND INTERMEDIATE VALUE THEOREM Claim.[17]
- To do this, well use the Squeeze theorem by establishing upper and lower bounds on sin(x)/x in an interval around 0.[18]
소스
- ↑ Squeeze theorem
- ↑ Brilliant Math & Science Wiki
- ↑ 이동: 3.0 3.1 3.2 3.3 Squeeze Theorem How-To w/ 4 Step-by-Step Examples!
- ↑ 이동: 4.0 4.1 4.2 Squeeze theorem
- ↑ Squeeze theorem
- ↑ How to use the squeeze theorem
- ↑ The Squeeze Theorem for Convergent Sequences
- ↑ Squeeze Theorem
- ↑ Determining Limits Using the Squeeze Theorem
- ↑ 이동: 10.0 10.1 10.2 Intuition Behind the Squeeze Theorem and Applications
- ↑ 이동: 11.0 11.1 When Is the Squeeze(Sandwich) Theorem Used?
- ↑ Squeeze theorem – Definition, Proof, and Examples
- ↑ 이동: 13.0 13.1 13.2 Undergraduate Mathematics/Squeeze theorem
- ↑ Squeeze Theorem: Definition and Examples - Math Class (Video)
- ↑ Squeezing Theorem -- from Wolfram MathWorld
- ↑ 이동: 16.0 16.1 Math 127 – calculus iii
- ↑ Using the squeeze theorem and intermediate
- ↑ The squeeze theorem: statement and example
메타데이터
위키데이터
- ID : Q1065257
Spacy 패턴 목록
- [{'LOWER': 'squeeze'}, {'LEMMA': 'theorem'}]
- [{'LOWER': 'pinching'}, {'LEMMA': 'theorem'}]
- [{'LOWER': 'sandwich'}, {'LEMMA': 'theorem'}]
- [{'LOWER': 'sandwich'}, {'LEMMA': 'rule'}]
- [{'LOWER': 'squeeze'}, {'LEMMA': 'lemma'}]