Power Series Assignment | College Homework Help

Power Series: What is a series? What are some tests used to determine whether an infinite series converges or diverges? What is a power series? What are some tests used to determine whether a power series converges on a particular domain? How can power series be applied to the study of differential equations? Given a series that converges how can a finite value be assigned to it in general? What is the intuitive motivation for Ces`aro summation? Are there series that are divergent when looking at the sequence of partial sums, but yet have a finite Ces`aro sum? (Basically just answers these questions, and related with “real-analysis” in Math) The goal of this assignment is for you to pick out a topic of your choice and explore the said topic. By design all material in mathematics usually has many theorems and results that can be discussed, but the goal is to pick out some themes and enough details so as to be able to provide enough information for a formal write-up. The write-up should contain: – a title – your name, the university’s name, and the due date – an abstract – an introduction – sections (the breakup of which completely depends on the topic chosen) – properly labeled lemmas, theorems, proofs, corollaries, examples, etc… – a conclusion – references Grade Breakup: The total is 100 points. 5 Points: Title, Name, University Name, Due Date, and Confirmation of Minimum Length of 7 Pages 10 Points: Abstract 10 Points: Introduction 60 Points: Main Content in Sections 10 Points: Conclusion 5 Points: ReferencesProject
MATH 105A
Due at 5pm on July 17, 2020
Overview: The goal of this assignment is for you to pick out a topic of your choosing and explore said topic. By design
all material in mathematics usually has many theorems and results that can be discussed, but the goal is to pick out some
themes and enough details so as to be able to provide enough information for a formal write-up.
Specifics:
The write-up is expected to be typed up using LATEX.
While the length of a write-up should never be the focus, some minimum has to be set. As such, the required
minimum of the write-up’s length is five pages.
The write-up should contain:
– a title
– your name, the university’s name, and the due date
– an abstract
– an introduction
– sections (the breakup of which completely depends on the topic chosen)
– properly labeled lemmas, theorems, proofs, corollaries, examples, etc…
– a conclusion
– references
Grade Breakup: The total is 100 points.
5 Points: Title, Name, University Name, Due Date, and Confirmation of Minimum Length of 7 Pages
10 Points: Abstract
10 Points: Introduction
60 Points: Main Content in Sections
10 Points: Conclusion
5 Points: References
Possible Topics: The following list gives some possible choices for you, though these are not the only ones allowed. If
you find material (related to real analysis) that you have an interest in that does not explicitly show up in the list below,
contact the instructor to approve the topic. Furthermore, once a topic has been chosen by ten people it cannot be chosen
by another. A piazza post will keep track of how many people have reserved which topic.
Classes of Functions: In the context of real functions there are collections Ck, which consist of functions that are
k-differentiable with k = 0 referencing continuous functions. What is the relation between Cn and Cm for distinct
m and n non-negative integers? Since differentiability is defined at a singular point, does there exist a function that
is continuous everywhere but not differentiable anywhere? Furthermore, does there exist a function that is nowhere
continuous? For the cases of smooth and analytic functions we typically use k = 1 and k = !, respectively. Are all
smooth functions analytic or vice versa? If one is more general determine a function which is one, but not the other.
Compare the breakdown of real functions, in terms of differentiability, to complex functions. Is there a significant
difference between the behavior observed in complex functions that does not occur in the real case?
Power Series: What is a series? What are some tests used to determine whether an infinite series converges or
diverges? What is a power series? What are some tests used to determine whether a power series converges on
a particular domain? How can power series be applied to the study of differential equations? Given a series that
converges how can a finite value be assigned to it in general? What is the intuitive motivation for Ces`aro summation?
Are there series that are divergent when looking at the sequence of partial sums, but yet have a finite Ces`aro sum?
Fourier Analysis: What is the historical context of Fourier analysis? What are some real world applications? How
are the Fourier series and Fourier transform defined? What are some symmetry relations involving a function and
its Fourier transform? What is the Gibbs phenomenon and is it simply a result of the imperfect measuring apparatus
used by human beings?
Lebesgue Measure: What is an abstract measure? On what spaces is the Lebesgue measure defined? What are
measurable sets and functions? How does the Lebesgue measure allow a reformulation of the Riemann integral? Is a
Lebesgue integral more general than a Riemann integral? What are Fatou’s lemma and the Dominated Convergence
theorem?
Topology: What is a topology? What is a topological space? What is a continuous function between two topological
spaces? How does a continuous function preserve the algebraic structure of a topological space? If there exists a
homeomorphism between two topological spaces do they necessarily have the same cardinality, are both connected,
are both compact, etc…? How is the definition of an abstract continuous function related to the – definition of
continuity over the real line (or even metric spaces)?