Infinite series are useful in mathematics and fields like physics, chemistry, biology, and engineering. They allow complicated functions to be expressed as the sum of infinitely many terms, which can then be directly solved. Fourier analysis breaks functions into infinite trigonometric series, allowing the study of wave phenomena. The area inside the Koch snowflake, a fractal shape generated by infinite recursion, can be found by summing an infinite geometric series. However, its perimeter is infinite because the length grows without bound at each step of recursion. Infinite series commonly arise in solving differential equations and representing functions and are applied in fields such as image compression, sound analysis, and electrical engineering.

1. INFINITE SERIES
• The sum of infinitely many numbers related in a given
way and listed in a given order. Infinite series are useful in
mathematics and in such disciplines as physics, chemistry,
biology, and engineering.
•Many mathematical problems that involve a complicated
function can be solved directly and easily when the function
can be expressed as an infinite series involving
trigonometric functions (sine and cosine).
•The process of breaking up a rather arbitrary function into
an infinite trigonometric series is called Fourier analysis or
harmonic analysis and has numerous applications in the
study of various wave phenomena.

2. In this we study an example that illustrates how infinite series arise in the
investigation of recursions and fractals, and why it is useful to know how
to treat them.
The Koch Snowflake: finite area but
infinite perimeter

3. The Koch snowflake is a geometric shape created by a repeated set of steps.
The shape itself is called a fractal, and has some remarkable properties. One of
these properties is "self -similarity". This refers to the fact that small parts of
the shape are very similar to the whole shape itself. The process of creating
this snowflake is, in principle, infinite, resulting in some peculiarities that we
will explore below.
• Steps in creating the Koch Snowflake:
• Start with an equilateral triangle

4. Add smaller copies of the triangle to
each side

5. Add yet smaller copies of the triangle
to each side again

6. Repeat the process...

7. Area contained inside the Koch
Snowflake:
• sum up the areas contained in each of the little triangles that we have added
• Let initial area of the triangle is A
• We first observe that the area of each of the pieces added at the next stage is 1/9
A
• This can be seen from the diagram below.
• Since three pieces are added, the area in increased by 3(1/9) = 1/3.
• The triangles at the next stage each contain 1/9 of the area of the triangle used in
the previous step, but now we add 4 of these per side, for a total of 3(4)=12
around the snowflake.
• At the next stage still, we will be adding 3(16) triangles with area again scaled by a
factor 1/9

8. Step 1:
Step 2:
Step 3:
Step 4:
Step n:
Continuing this process for increasing n leads to a series whose terms are of the form
The series above is just a simple geometric series with the base 4/9, and we know that
since this number is less than 1, the series converges. Indeed,

9. Result
• Thus, we found that the area inside the Koch
snowflake is finite, and that it sums up to
eight fifths of the area in the original triangle.
Since the shape is generated by an infinite
recursion (repetition of "the same" geometric
manipulation over and over again), we had to
sum an infinite series to obtain our result.

10. Perimeter of the Koch Snowflake:
• To simplify the matter, let us describe
what happens to one side of the triangle
as the recursion is repeated. Suppose that
the original length of one side is L.
• Now we go through the following steps:

11. Step 1:
Step 2:
Step 3:
Step n:
• The lengths are not added in this case.
• Rather, at each stage of the process, the length of one of the
original sides of the triangle is expanded by a factor of (4/3).
• This, as we have seen, leads to a sequence (not a series, just a
list of numbers) of the form L=(4/3)n
• But this sequence does not converge. Its terms are powers of a
number greater than 1, and thus these terms grow without
bound.
1 segment of length L.
4 segments, each of length L/3.
Total length of side is (4/3)L.
4 times 4 segments, each of length (1/3) L/3.
Total length of side is (4/3)(4/3) = (4/3)2.
Total Length =(4/3)n

12. Result
• The perimeter of the Koch snowflake is
infinite, even though its area is finite.
• We have solved an Infinite series for finding
the AREA of Koch Snowflake
• We have solved a Sequence for finding the
Perimeter of the Koch Snowflake.

13. One of the best applications of infinite series is
In harmonic analysis
Any periodic function can be expressed as an infinite
series of sine and cosine functions (given that
appropriate conditions are satisfied). This is used to
then analyse the original periodic function, and then
apply filters to it.
For example, a sound recording can have its bass
removed, or amplified, using roughly this technique.

14. • Some differential equations cannot be solved
using just one function, but can be
approximated as an infinite series (of powers of
x). This method is used in the Taylor series
expansion method and in the Frobenius
method (where the answer is assumed to be a
Frobenius series).

15. A simple application is to convert a fraction into a
recurring decimal, which is basically an infinite series.
For example, 1/3 = 0.3333333..., which can be written as
3/10 + 3/100 + 3/1000 + 3/10000 + ...
Hence we have
3/10 + 3/100 + 3/1000 + 3/10000 + ... = 1/3
which means that the more fractions you sum up from
the L.H.S., the closer you get to 1/3. Actually
0.3333333... = 1/3 means essentially the same thing; the
more digits you write in the decimal representation of
1/3, the closer you're getting to 1/3.

16. • Many mathematical problems that involve a
complicated function can be solved directly
and easily when the function can be expressed
as an infinite series involving trigonometric
functions (sine and cosine). The process of
breaking up a rather arbitrary function into an
infinite trigonometric series is called Fourier
analysis or harmonic analysis and has
numerous applications in the study of various
wave phenomena.

17. • Fourier series, In mathematics, an infinite series
used to solve special types of differential
equations. It consists of an infinite sum of sines
and cosines, and because it is periodic (i.e., its
values repeat over fixed intervals), it is a useful
tool in analyzing periodic functions. Though
investigated by Leonhard Euler, among others,
the idea was named for Joseph Fourier, who fully
explored its consequences, including important
applications in engineering, particularly in heat
conduction.

18. • Take for example JPEG image compression. The changing pattern of colours in an
image can be fitted by an fourier series (in practise it is a cosine series that is used)
As an infinite series could take an infinite amount of information to store it, that
doesn't seem like a good thing, but the infinite series can be approximated by the
first few terms. That means that instead of keeping the image in memory you only
need keep the first few terms of an infinte series - a big saving in memory. (details
in source)
• This is a common example - You don't actually use the infinite series when you
make a jpeg, but the people who invented jpeg couldn't have done so without
understanding series.
• In general applications of fourier series are widespread in engineering. They are
used in the analysis of current flow in electrical engineering. They are used analysis
of sound waves. They are used in mathematics to solve differential equations.
Fourier’s ideas can also be found in electronically synthesized music and talking
computer chips.
• https://www.rose-hulman.edu/~bryan/invprobs/jpegtalk2.pdf
[Picture Perfect: The mathematics of JPEG compression, Kurt Bryan, May 19, 2011]

19. • Sine and cosine are series; infinite series. Can
you do without them?
Fourier transforms, Laplace transforms,
spherical and ellipsoidal harmonics,
exponential and logarithmic functions, and
elliptic functions are all based on infinite
series. Can you do without those?

20. • yes, many infact ... in physics
example, bouncing of a ball!!!...if we have to
find out the time it takes to come to rest, its
an infinite converging series!!!...
to find the limits of various functions ...

21. Convergence is when two or more things come together to form a new
whole, like the convergence of plum and apricot genes in the plucot.
Convergence comes from the
prefix Con- together
the verb verge - to turn toward.
convergence → to describe things that are in the process of
coming together.
Ex: the slow convergence of your opinions with those of your
mother.
for things that have already come together, like the
convergence of two roads, or for the place where two
things already overlap, like the convergence of your aunt's
crazy wardrobe with avant-garde fashion
Mathematics:
The property or manner of approaching a limit, such as a
point, line, function, or value is called convergence.

22. CONVERGENCE
• A series is convergent if the sequence of its
partial sums tends to a limit; that means that
the partial sums become closer and closer to a
given number when the number of their
terms increases.

23. What is a SEQUENCE?
It is a mapping whose
domain is the set of
Natural Numbers.

24. 12. 1 A sequence is…
(a) an ordered list of objects.
(b) A function whose domain is a set of natural numbers.
Domain: 1, 2, 3, 4, …,n…
Range a1, a2, a3, a4, … an…
1 1 1 1
1, , , , ...
2 4 8 16
{(1, 1), (2, ½), (3, ¼), (4, 1/8) ….}

25. MSP SI 2007 Sequences
Number Patterns
Find the next two terms of each sequence.
Describe how you found each term.
0, 1, 3, 6, 10, 15, ___, ___
11, 22, 33, 44, 55, ___, ___
5, 8, 7, 10, 9, 12, 11, __,__
66 77
21 28
14 13
Slide Courtesy of Guy Barmoha

26. MSP SI 2007 Sequences
Sequences

(a,ar1
,ar2
,ar3
...)

F(n) :
0
1
F(n 1)  F(n  2)










Examples Sequence Notation

(a1,a2,a3,...an )

(1,2,3,...)

(1,4,7,...)

arithmetic

(2,4,8,...)

geometric

(1,1,2,3,5,8...)

Fibonacci

(1,
1
2
,
1
4
,
1
8
,...)

27. MSP SI 2007 Sequences
Arithmetic Sequences
Sequence of numbers where any 2
successive members have a common
difference
Example:
( 0, 1, 2, 3, 4 )
+ 1 +1 +1 +1

28. MSP SI 2007 Sequences
Arithmetic Sequences
Sequence of numbers where any 2
successive members have a common
difference
Example:
( 0, 3, 6, 9, 12 )
+ 3 +3 +3 +3

29. Infinite Series
The sum of the terms of an infinite sequence is called an
infinite series.
Notation:
1 2 3 4
1
...k
k
a a a a a


    
NOTES:
• ak is some function of k whose domain is a set of integers.
• k can start anywhere (0 or 1 is the most common)
• The following are all the same:
1 1
k n n
k n
a a a
 
 
   

30. Partial Sums of an Infinite Series
1 2 3 4
1
...k
k
a a a a a


    
2 1 2s a a 
3 1 2 3s a a a  
1 2 3
1
...
n
n n k
k
s a a a a a

      
1 1s a
  1n n
s


Sequence of
partial sums.
●
●
●
Recursive Definition:
1n n ns s a 

31. Converging/Diverging Series
1 2 3 4
1
...k
k
a a a a a


    
  1n n
s


If converges to ,S
then the series converges and
1
k
k
a S



If the sequence of partial sums diverges, then so
does the series (it has no sum).
S is not often easy or even possible to determine!

32. Example …
1
1 1 1 1 1
...
2 2 4 8 16k
k


    
1
1
2
s 
2
1 1 3
2 4 4
s   
3
1 1 1 3 1 7
2 4 8 4 8 8
s      
4
1 1 1 1 7 1 15
2 4 8 16 8 16 16
s       
Pattern?
2 1
2
n
n n
s


2 1
lim
2
n
nn

 1
1
lim 1
2nn
 
  
 
1
NOTE: A general expression for sn is usually difficult to determine.

33. Geometric Series
Each term is obtained by multiplying the proceeding term by a
fixed constant.
1 2 3
1
...k
k
ar a ar ar ar



    
Example:
1
3 3 3 3
...
10 10 100 1000k
k


   
NOTE: w/ geometric series, k can start with any value (usually 0 or 1).
3
10
a 
1
10
r 
0
k
k
ar


 

34. Geometric Series
1
1 0
k k
k k
ar ar
 

 
 
• a is the value of the first term
• r is the “common ratio”
• r > 0, all terms have the same sign
• r < 0, terms alternate signs

35. Repeating decimals-Geometric
Series
2 4 6 8
8 8 8 8
0.080808 ...
10 10 10 10
    
21
1
2
8
810
11 991
10
n
n
a
ar
r



  
 

The repeating decimal is equivalent to 8/99.
2 2
8 1
10 10
a and r 
1
1
2 2
1 1
8 1
10 10
n
n
n n
ar
 

 
 
  
 
 

36. THANKS