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### Analysis.pptx

• 1. Analysis - Dewang Shukla - Vaibhav Khatri - Abhishek Chouhan - Aryan Verma Mathematics is full of surprises, see how easy it is to prove that 2=1 Wow! this is why you fail to compromise the citizens and congress is not able to form government. You really need to study the basics of mathematics. Just a joke
• 2. ‘Σ’Group Dewang Shukla G1 Vaibhav Khatri G2 Aryan Verma G3 Abhishek Chouhan G4 2 Analysis 2023
• 3. Index 1 2 3 4 5 Introduction Why do Analysis Defining Natural Numbers 0 & ∞, Series, L'Hôpital's rule Paradox 3 Analysis 2023 6 Summary
• 4. Introduction 4 Today, we will be discussing some fundamental rules and do an exercise based on basic analysis involving proofs. We will be also answering some interesting questions and discussing a paradox based on our research on 'series'. Analysis is the rigorous study of a particular topic starting from the very basic and getting valuable results in the process. With Math
• 6. ‘Division By Zero’ 6 Analysis 2023 Example 1- We are very familiar with the cancellation law, 𝑎𝑐 = 𝑏𝑐 ⇒ 𝑎 = 𝑏 does not work when c = 0. For instance, the identity , 1 x 0 = 2 x 0 is true, but if one blindly cancels the 0 then we will obtain 1 = 2 ,which is false.
• 7. ‘Divergent Series’ 7 Analysis 2023 Example 2- We have seen geometric series such as infinite sum, 𝑆 = 1 + 1 2 + 1 4 + 1 8 + 1 16 + ⋯ … … … … 2𝑆 = 2 + 1 + 1 2 + 1 4 + 1 8 + 1 16 + ⋯ … … … … = 2 + 𝑆 So, 𝑆 = 2 However, on applying this on a series like 𝑆 = 1 + 2 + 4 + 8 + 16 + ⋯ … … … … We will get, 2𝑆 = 2 + 4 + 8 + 16 + ⋯ … … … … = 𝑆 − 1 Or, 𝑆 = −1 which seems absurd. A similar example is, 𝑆 = 1 − 1 + 1 − 1 + 1 − 1 + 1 − 1 + ⋯ … … … 𝑆 = 1 − 1 − 1 + 1 − 1 + 1 − 1 + 1 − 1 + 1 − 1 + ⋯ … … = 1 − 𝑆 So, 𝑆 = 1 2 But as we think about it we should get the answer 0. So, which one is correct?
• 8. Interchanging Sums 8 Analysis 2023 Example 3- Consider any matrix of numbers. For example, 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 On doing the sums of all the rows and the sums of all the columns, then total all the rows sums and column sums, we get the same number, This is in fact a property for matrices. Now consider an infinite matrix, 1 0 0 −1 1 0 0 −1 1 0 … 0 … 0 … 0 0 −1 ∶ ∶ ∶ 1 … : : On applying the same logic as we did above we will get total sums of all the rows with other rows as 1, but for column it will be zero !! So does this mean that the property of interchanging sums is false? 6 15 24 18 15 12 45
• 9. L'Hôpital's rule 9 Analysis 2023 Example 4- We are familiar with the beautifully simple L'Hôpital's rule. lim 𝑥→𝑥0 𝑓 𝑥 𝑔 𝑥 = lim 𝑥→𝑥0 𝑓′ 𝑥 𝑔′ 𝑥 Now, let f(x) := x, g(x) := 1+x and 𝑥0 = 0,then by the above rule we will get, lim 𝑥→0 𝑥 1 + 𝑥 = lim 𝑥→0 1 1 = 1, But this is incorrect as we can clearly see the answer would be 0 by directly putting x=0. Another example, lim 𝑥→0 𝑥2sin(𝑥−4) 𝑥 On the applying the rule we will obtain lim 𝑥→0 2𝑥𝑠𝑖𝑛(𝑥−4) − lim 4𝑥−3cos(𝑥−4) On putting the value of x as 0 in the above equation we will obtain an answer tending to infinity, 𝑥−3 → ∞ Hence L'Hôpital's rule is quite rigorous to use.
• 10. Why is '3' a natural number? 20XX Presentation title 10
• 11. Defining Natural Numbers (Using Peano Axioms) Zero is a natural number. Zero is not the successor of any natural number. Principle of mathematical induction Every natural number has a successor in the natural numbers. If the successor of two natural numbers is the same, then the two original numbers are the same.
• 12. 12 Analysis 2023 Definition1. (Informal) A natural number is any element of the set N = {0,1,2,3,4, ... } To define the natural numbers, we will use two fundamental concepts: The zero number 0, and the increment operation. In deference to modern computer languages, we will use n++ to denote the increment or successor of n, thus for instance 3++ = 4. So, it seems like we want to say that N consists of 0 and everything which can be obtained from 0 by incrementing: N should consist of the objects O,O++,(O++)++,((O++)++)++,etc.
• 13. Axiom 1. 0 is a natural number. Axiom 2. If n is a natural number, then n++ is also a natural number. Definition 2. We define 1 to be the number 0++, 2 to be the number (0++ )++, 3 to be the number ((0++ )++)++,etc. (In other words, 1 := 0++, 2 := 1 ++, 3 := 2++, etc. In this text We use "x := y" to denote the statement that x is defined to equal y.) Axiom 3. 0 is not the successor of any natural number; i.e., we have n++ # 0 for every natural number n. Axiom 4. Different natural numbers must have different successors; i.e., if n, m are natural numbers and n is not equal to m, then n++ is not equal to m++. Equivalently, if n++ = m ++ , then we must have n := m. 13 Analysis 2023
• 14. 14 Analysis 2023 Axiom 5. (Principle of mathematical induction). Let P(n) be any property pertaining to a natural number n. Suppose that P(O) is true and suppose that whenever P(n) is true, P(n++) is also true. Then P( n) is true for every natural number n. This axiom confirms that we only have numbers like 0,1,2,3,etc. and not like 0.5 or other rational. Axioms 1-5 are known as the Peano axioms for the natural numbers. They are all very plausible, and so we shall make Assumption: There exists a number system N, whose elements we will call natural numbers, for which Axioms 1-5 are true.
• 15. Natural numbers can approach ‘∞′ , but never actually reach it. ∞ is not one of the natural numbers. 2023 Analysis 15
• 17. Zero & Infinity a = b a2 = ab a2 – b2 = ab – b2 (a + b)(a – b) = b(a – b) a + b = b b + b = b 2b = b 2 = 1 17 Analysis 2023 So, we have to take care that we do not prove such absurd equations by handling 0 and ∞ properly
• 18. Divergent and convergent Series A series that gives a non-singular or not defined answer is divergent while a series giving a finite answer is convergent. We have already mentioned several divergent series and saw that how we were getting absurd answers in the process. So, we say ''We should not perform operations on Divergent Series.'' For instance in the example of infinite matrix, the sum of the rows and columns gave us a divergent series, and for correctness, we say, ''Sums can only be interchanged for a sum that is absolutely convergent.'' 18 Analysis 2023
• 19. L'Hôpital's rule This rule is a general method used to evaluate the limits of intermediate forms such as 𝟎 𝟎 or ∞ ∞ . It says that, 𝒍𝒊𝒎 𝒙→𝒙𝟎 𝒇 𝒙 𝒈 𝒙 = 𝒍𝒊𝒎 𝒙→𝒙𝟎 𝒇′ 𝒙 𝒈′ 𝒙 However, we cannot blindly apply it anywhere.  Given f and g are two functions such that f and g are both differentiable at a point c. Also f(c)=g(c)=0 and g'(c)≠0 only then we can have, 𝒍𝒊𝒎 𝒙→𝒄 𝒇 𝒙 𝒈 𝒙 = 𝒍𝒊𝒎 𝒙→𝒄 𝒇′ 𝒙 𝒈′ 𝒙 So while applying this rule, we have to check our functions first. 19 Analysis 2023
• 20. Ramanujan’s Paradox 1 + 2 + 3 + 4 + ⋯ … … … … = − 1 12 Yes, it really seems absurd, but such is the nature of a paradox. 20 Analysis 2023 So far, we talked about how operations on divergent series may lead to complicated answers and we should try to avoid them, but if a divergent series really exists then there must be some way to simplify and represent them. Many mathematicians have given different theories in form of a 'paradox'. We have included the famous Ramanujan's Summation according to which the sum of natural numbers gives us a finite negative number.
• 21. Thank you Analysis I – Terence Tao Special thanks to ‘Dr.Satish Shukla’
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