The document discusses sequences and their properties. A sequence is a function whose domain is the positive integers. Sequences are commonly represented using subscript notation rather than standard function notation. The nth term of a sequence is denoted an. [/SUMMARY]
ppt on Vector spaces (VCLA) by dhrumil patel and harshid panchalharshid panchal
this is the ppt on vector spaces of linear algebra and vector calculus (VCLA)
contents :
Real Vector Spaces
Sub Spaces
Linear combination
Linear independence
Span Of Set Of Vectors
Basis
Dimension
Row Space, Column Space, Null Space
Rank And Nullity
Coordinate and change of basis
this is made by dhrumil patel which is in chemical branch in ld college of engineering (2014-18)
i think he is the best ppt maker,dhrumil patel,harshid panchal
ppt on Vector spaces (VCLA) by dhrumil patel and harshid panchalharshid panchal
this is the ppt on vector spaces of linear algebra and vector calculus (VCLA)
contents :
Real Vector Spaces
Sub Spaces
Linear combination
Linear independence
Span Of Set Of Vectors
Basis
Dimension
Row Space, Column Space, Null Space
Rank And Nullity
Coordinate and change of basis
this is made by dhrumil patel which is in chemical branch in ld college of engineering (2014-18)
i think he is the best ppt maker,dhrumil patel,harshid panchal
Sets & Set Operation
CMSC 56 | Discrete Mathematical Structure for Computer Science
September 11, 2018
Instructor: Allyn Joy D. Calcaben
College of Arts & Sciences
University of the Philippines Visayas
For more instructional resources, CLICK me here and DON'T FORGET TO SUBSCRIBE!
https://tinyurl.com/y9muob6q
LIKE and FOLLOW me here!
https://tinyurl.com/ycjp8r7u
https://tinyurl.com/ybo27k2u
Series solution of ordinary differential equation
advance engineering mathematics
The power series method is the standard method for solving linear ODEs with variable
coefficients. It gives solutions in the form of power series. These series can be used for computing values, graphing curves, proving formulas, and exploring properties of
solutions, as we shall see.
For a system involving two variables (x and y), each linear equation determines a line on the xy-plane. Because a solution to a linear system must satisfy all of the equations, the solution set is the intersection of these lines, and is hence either a line, a single point, or the empty set
Sets & Set Operation
CMSC 56 | Discrete Mathematical Structure for Computer Science
September 11, 2018
Instructor: Allyn Joy D. Calcaben
College of Arts & Sciences
University of the Philippines Visayas
For more instructional resources, CLICK me here and DON'T FORGET TO SUBSCRIBE!
https://tinyurl.com/y9muob6q
LIKE and FOLLOW me here!
https://tinyurl.com/ycjp8r7u
https://tinyurl.com/ybo27k2u
Series solution of ordinary differential equation
advance engineering mathematics
The power series method is the standard method for solving linear ODEs with variable
coefficients. It gives solutions in the form of power series. These series can be used for computing values, graphing curves, proving formulas, and exploring properties of
solutions, as we shall see.
For a system involving two variables (x and y), each linear equation determines a line on the xy-plane. Because a solution to a linear system must satisfy all of the equations, the solution set is the intersection of these lines, and is hence either a line, a single point, or the empty set
An alternative scheme for approximating a periodic functionijscmcj
Fourier series is generally used in applied mathematics and non-linear mechanics to solve the problems containing periodic functions. The aim of this paper is to present a new scheme through involving the Taylor’s series after some process modification for all such problems. It will save the long evaluation process involve in computing the different constants of Fourier series. The result obtained by the present
method has been compared with the result from the Fourier series expansion w.r.t. the exact solution and
found to be more satisfactory.
Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
Exploiting Artificial Intelligence for Empowering Researchers and Faculty, In...Dr. Vinod Kumar Kanvaria
Exploiting Artificial Intelligence for Empowering Researchers and Faculty,
International FDP on Fundamentals of Research in Social Sciences
at Integral University, Lucknow, 06.06.2024
By Dr. Vinod Kumar Kanvaria
Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
Safalta Digital marketing institute in Noida, provide complete applications that encompass a huge range of virtual advertising and marketing additives, which includes search engine optimization, virtual communication advertising, pay-per-click on marketing, content material advertising, internet analytics, and greater. These university courses are designed for students who possess a comprehensive understanding of virtual marketing strategies and attributes.Safalta Digital Marketing Institute in Noida is a first choice for young individuals or students who are looking to start their careers in the field of digital advertising. The institute gives specialized courses designed and certification.
for beginners, providing thorough training in areas such as SEO, digital communication marketing, and PPC training in Noida. After finishing the program, students receive the certifications recognised by top different universitie, setting a strong foundation for a successful career in digital marketing.
Unit 8 - Information and Communication Technology (Paper I).pdfThiyagu K
This slides describes the basic concepts of ICT, basics of Email, Emerging Technology and Digital Initiatives in Education. This presentations aligns with the UGC Paper I syllabus.
Embracing GenAI - A Strategic ImperativePeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
4. Sequences
A sequence is defined as a function whose domain is the
set of positive integers. Although a sequence is a function,
it is common to represent sequences by subscript notation
rather than by the standard function notation. For instance,
in the sequence
Sequence
1 is mapped onto a1, 2 is mapped onto a2, and so on. The
numbers a1, a2, a3, . . ., an, . . . are the terms of the
sequence. The number an is the nth term of the sequence,
and the entire sequence is denoted by {an}.
4
5. Example 1 – Listing the Terms of a Sequence
a. The terms of the sequence {an} = {3 + (–1)n} are
3 + (–1)1, 3 + (–1)2, 3 + (–1)3, 3 + (–1)4, . . .
2,
4,
2,
4,
....
b. The terms of the sequence {bn}
are
5
6. Example 1 – Listing the Terms of a Sequence
c. The terms of the sequence {cn}
cont’d
are
d. The terms of the recursively defined sequence {dn},
where d1 = 25 and dn + 1 = dn – 5, are
25, 25 – 5 = 20, 20 – 5 = 15, 15 – 5 = 10,. . . . .
6
8. Limit of a Sequence
Sequences whose terms approach to limiting values, are
said to converge. For instance, the sequence {1/2n}
converges to 0, as indicated in the following definition.
8
9. Limit of a Sequence
Graphically, this definition says that eventually
(for n > M and ε > 0) the terms of a sequence that
converges to L will lie within the band between the lines
y = L + ε and y = L – ε as shown in Figure 1.
If a sequence {an} agrees with a
function f at every positive integer,
and if f(x) approaches a
limit L as
the sequence must
converge to the same limit L.
For n > M the terms of the sequence
all lie within ε units of L.
Figure 1
9
11. Example 2 – Finding the Limit of a Sequence
Find the limit of the sequence whose nth term is
Solution:
You learned that
So, you can apply Theorem 1 to conclude that
11
13. Limit of a Sequence
The symbol n! (read “n factorial”) is used to simplify some
of the formulas. Let n be a positive integer; then n factorial
is defined as n! = 1 • 2 • 3 • 4 . . . (n – 1) • n.
As a special case, zero factorial is defined as 0! = 1.
From this definition, you can see that 1! = 1, 2! = 1 • 2 = 2,
3! = 1 • 2 • 3 = 6, and so on.
13
14. Example 5 – Using the Squeeze Theorem
Show that the sequence
find its limit.
converges, and
Solution:
To apply the Squeeze Theorem, you must find two
convergent sequences that can be related to the given
sequence.
Two possibilities are an = –1/2n and bn = 1/2n, both of which
converge to 0.
By comparing the term n! with 2n, you can see that,
n! = 1 • 2 • 3 • 4 • 5 • 6 . . . n =
and
2n = 2 • 2 • 2 • 2 • 2 • 2 . . . 2 =
14
15. Example 5 – Solution
cont’d
This implies that for n ≥ 4, 2n < n!, and you have
as shown in Figure 2.
So, by the Squeeze Theorem
it follows that
For n ≥ 4, (–1)n /n! is squeezed between
–1/2n and 1/2n.
Figure 2
15
18. Pattern Recognition for Sequences
Sometimes the terms of a sequence are generated by
some rule that does not explicitly identify the nth term of the
sequence.
In such cases, you may be required to discover a pattern in
the sequence and to describe the nth term.
Once the nth term has been specified, you can investigate
the convergence or divergence of the sequence.
18
19. Example 6 – Finding the nth Term of a Sequence
Find a sequence {an} whose first five terms are
and then determine whether the particular sequence you
have chosen converges or diverges.
19
20. Example 6 – Solution
First, note that the numerators are successive powers of 2,
and the denominators form the sequence of positive odd
integers.
By comparing an with n, you have the following pattern.
Using L’Hôpital’s Rule to evaluate the limit of
f(x) = 2x/(2x – 1), you obtain
So, the sequence diverges.
20
23. Example 8 – Determining Whether a Sequence Is Monotonic
Determine whether each sequence having the given nth
term is monotonic.
Solution:
a. This sequence alternates between
2 and 4.
So, it is not monotonic.
Not monotonic
Figure 3(a)
23
24. Example 8 – Solution
cont’d
b. This sequence is monotonic because each successive
term is larger than its predecessor.
To see this, compare the terms
bn and bn + 1.
Monotonic
Figure 3(b)
24
25. Example 8 – Solution
cont’d
c. This sequence is not monotonic, because the second
term is larger than the first term, and larger than the
third.
(Note that if you drop the first term,
the remaining sequence c2, c3, c4, . . .
is monotonic.)
Not monotonic
Figure 3(c)
25
27. Monotonic Sequences and Bounded Sequences
One important property of the real numbers is that they are
complete. This means that there are no holes or gaps on
the real number line. (The set of rational numbers does not
have the completeness property.)
The completeness axiom for real numbers can be used to
conclude that if a sequence has an upper bound, it must
have a least upper bound (an upper bound that is smaller
than all other upper bounds for the sequence).
27
28. Monotonic Sequences and Bounded Sequences
For example, the least upper bound of the sequence
{an} = {n/(n + 1)},
is 1.
28
29. Example 9 – Bounded and Monotonic Sequences
a. The sequence {an} = {1/n} is both bounded and
monotonic and so, by Theorem 5, must converge.
b. The divergent sequence {bn} = {n2/(n + 1)} is monotonic,
but not bounded. (It is bounded below.)
c. The divergent sequence {cn} = {(–1)n} is bounded, but
not monotonic.
29