"샌드위치 정리"의 두 판 사이의 차이

수학노트
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(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> ...)
 
 
(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>  
$a_n$은 다음 점화식을 만족시킨다 $$a_0=\pi$$ $$a_1=2$$ $$a_{n}=\frac{n-1}{n}a_{n-2} \label{rec}$$
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<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 $$\lim_{n\to \infty } \, \frac{a_{2 n}}{a_{2 n+1}}=1 \label{lim}$$
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다음과 같은 극한을 계산하는 문제 :<math>\lim_{n\to \infty } \, \frac{a_{2 n}}{a_{2 n+1}}=1 \label{lim}</math>
  
  
$a_{n}$은 단조감소수열이므로, 다음 부등식이 성립한다
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<math>a_{n}</math>은 단조감소수열이므로, 다음 부등식이 성립한다
$$1 \le \frac{a_{2n}}{a_{2n+1}} \le \frac{a_{2n-1}}{a_{2n+1}}=\frac{2n+1}{2n}$$
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:<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 $$\lim_{n\to \infty } \, \frac{a_{2 n}}{a_{2 n+1}}=1$$
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따라서 [[샌드위치 정리]]에 의해 :<math>\lim_{n\to \infty } \, \frac{a_{2 n}}{a_{2 n+1}}=1</math>
 +
 
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== 노트 ==
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===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>
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===소스===
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<references />
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== 메타데이터 ==
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===위키데이터===
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* ID :  [https://www.wikidata.org/wiki/Q1065257 Q1065257]
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===Spacy 패턴 목록===
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* [{'LOWER': 'squeeze'}, {'LEMMA': 'theorem'}]
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* [{'LOWER': 'pinching'}, {'LEMMA': 'theorem'}]
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* [{'LOWER': 'sandwich'}, {'LEMMA': 'theorem'}]
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* [{'LOWER': 'sandwich'}, {'LEMMA': 'rule'}]
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* [{'LOWER': 'squeeze'}, {'LEMMA': 'lemma'}]

2021년 2월 26일 (금) 01:29 기준 최신판

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

  1. The squeeze theorem is used in calculus and mathematical analysis.[1]
  2. 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]
  3. How to use the squeeze theorem?[3]
  4. In fact, that’s the whole idea behind the squeeze theorem, also known as the pinching theorem or the sandwich theorem.[3]
  5. But with the help of the squeeze theorem, we can now determine the limit of an oscillating function![3]
  6. 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]
  7. This exercise explores the squeeze or sandwich theorem.[4]
  8. The user is expected use the function to determine the limit via the squeeze theorem and provide the correct answer.[4]
  9. The squeeze theorem is also sometimes called the sandwich theorem.[4]
  10. The squeeze theorem is another way to solve for tricky limits.[5]
  11. 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]
  12. 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]
  13. This result is also known, in the UK in particular, as the sandwich theorem or the sandwich rule.[8]
  14. 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]
  15. The squeeze theorem espresses in precise mathematical terms a simple idea.[10]
  16. You start to see how we'll use the squeeze theorem?[10]
  17. Now, we're ready to use the squeeze theorem![10]
  18. When is the squeeze theorem applied?[11]
  19. Are there other such functions to apply when using the squeeze theorem?[11]
  20. 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]
  21. 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]
  22. The squeeze theorem is a technical result that is very important in proofs in calculus and mathematical analysis.[13]
  23. The squeeze theorem is formally stated as follows.[13]
  24. Some people call it the sandwich theorem, but I like the term squeeze.[14]
  25. The squeezing theorem is also called the sandwich theorem.[15]
  26. Can we apply Squeeze theorem for the following limits?[16]
  27. If a two variable function satisfy the requirements, then we may apply squeeze theorem.[16]
  28. 1 2 USING THE SQUEEZE THEOREM AND INTERMEDIATE VALUE THEOREM Claim.[17]
  29. 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

메타데이터

위키데이터

Spacy 패턴 목록

  • [{'LOWER': 'squeeze'}, {'LEMMA': 'theorem'}]
  • [{'LOWER': 'pinching'}, {'LEMMA': 'theorem'}]
  • [{'LOWER': 'sandwich'}, {'LEMMA': 'theorem'}]
  • [{'LOWER': 'sandwich'}, {'LEMMA': 'rule'}]
  • [{'LOWER': 'squeeze'}, {'LEMMA': 'lemma'}]