샌드위치 정리

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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

  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

  1. ↑ Squeeze theorem
  2. ↑ Brilliant Math & Science Wiki
  3. ↑ 이동: 3.0 3.1 3.2 3.3 Squeeze Theorem How-To w/ 4 Step-by-Step Examples!
  4. ↑ 이동: 4.0 4.1 4.2 Squeeze theorem
  5. ↑ Squeeze theorem
  6. ↑ How to use the squeeze theorem
  7. ↑ The Squeeze Theorem for Convergent Sequences
  8. ↑ Squeeze Theorem
  9. ↑ Determining Limits Using the Squeeze Theorem
  10. ↑ 이동: 10.0 10.1 10.2 Intuition Behind the Squeeze Theorem and Applications
  11. ↑ 이동: 11.0 11.1 When Is the Squeeze(Sandwich) Theorem Used?
  12. ↑ Squeeze theorem – Definition, Proof, and Examples
  13. ↑ 이동: 13.0 13.1 13.2 Undergraduate Mathematics/Squeeze theorem
  14. ↑ Squeeze Theorem: Definition and Examples - Math Class (Video)
  15. ↑ Squeezing Theorem -- from Wolfram MathWorld
  16. ↑ 이동: 16.0 16.1 Math 127 – calculus iii
  17. ↑ Using the squeeze theorem and intermediate
  18. ↑ The squeeze theorem: statement and example

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Spacy 패턴 목록

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