- The document discusses some fundamental concepts in analysis such as natural numbers, zero, infinity, series, and L'Hôpital's rule. - It provides examples to illustrate issues that can arise from improperly handling concepts like division by zero, divergent series, and interchange of sums. - The key points are that operations should not be performed on divergent series and sums can only be interchanged if the series is absolutely convergent. L'Hôpital's rule also has specific requirements that must be checked before applying it.

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

5. Why do
Analysis?
‘Knowing the rules
properly’

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

16. Some Basic
Rules
Solutions to the
examples

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’