Numbers Sequences and Series

Revision Guide

Author
Affiliation

Dr. Silvio Fanzon

University of Hull

Published

6 Dec 2024

Revision Guide

Revision Guide for the Exam of the module Numbers Sequences and Series 400297 2024/25 at the University of Hull. If you have any question or find any typo, please email me at

S.Fanzon@hull.ac.uk

Full lenght Lecture Notes of the module available at

silviofanzon.com/2024-NSS-Notes

Checklist

You should be comfortable with the following topics/taks:

Preliminaries

  • Prove that \(\sqrt{p} \notin \mathbb{Q}\) for \(p\) a prime number
  • Compute infinite union / intersection
  • Show that a binary relation is of equivalence / order / total order
  • Characterize the equivalence classes of a given equivalence relation
  • Prove statements by induction (such as Bernoulli’s inequality)
  • Compute the absolute value of a real number
  • Understand how to apply triangle inequality

Real Numbers

  • Determine if a given set with binary operation is a field
  • Prove uniqueness of neutral element / inverse
  • Computing Sup / Max and Inf / Min of a given set
  • Prove that a given set is inductive
  • Remember that \(\mathbb{N}\), \(\mathbb{Z}\) are not fields, \(\mathbb{Q}\) is an ordered field, \(\mathbb{R}\) is a complete ordered field
  • State the axiom of completeness

Properties of \(\mathbb{R}\)

  • Know how to use the Archimedean property
  • Characterization of sup / inf in terms of \(\varepsilon\)
  • Sup / Inf and Max / Min of intervals
  • Determine if a given set is finite / countable / uncountable
  • Remember that \(\mathbb{N}\), \(\mathbb{Z}\), \(\mathbb{Q}\) are countable
  • Remember that \(\mathbb{R}\) and the irrationals are uncountable

Complex Numbers

  • Sum, multiplication, division, conjugate of complex numbers
  • Computing the inverse of a complex number
  • Find modulus and argument of a complex number
  • Compute Cartesian, Trigonometric and Exponential form of a complex number
  • Complex exponential and its properties
  • Computing powers of complex numbers
  • Solving degree 2 polynomial equations in \(\mathbb{C}\)
  • Long division of polynomials
  • Solving higher degree polynomial equations in \(\mathbb{C}\)
  • Finding the roots of unity
  • Finding the n-th roots of a complex number

Sequences in \(\mathbb{R}\)

  • Use the definition of convergence to prove convergence of a given real sequence
  • Prove that a given sequence is bounded
  • Remember that convergent sequences are bounded
  • Use the Algebra of Limits to prove convergence / divergence of a given sequence
  • Use the Squeeze Theorem to prove convergence of a given sequence
  • Use the Geometric Sequence Test to prove convergence / divergence of a given sequence
  • Use the Ratio Test to prove convergence / divergence of a given sequence
  • Prove that a sequence is monotone increasing / decreasing
  • Know the statement of the Monotone Convergence Theorem
  • Memorize the 4 Special Limits, and know how to apply them to study convergence / divergence of a given sequence

Sequences in \(\mathbb{C}\)

  • Use the definition of convergence to prove convergence of a given complex sequence
  • Prove that a complex sequence is bounded
  • Use the Algebra of Limits to prove convergence / divergence of a given sequence
  • Use the Geometric Sequence Test / Ratio Test to prove convergence / divergence of a given complex sequence
  • Determine convergence of real and imaginary part of a given complex sequence

Series

  • Compute the partial sums of a given series
  • Compute the sum of a telescopic series
  • Apply the Necessary Condition for Convergence to prove that a given series is divergent
  • Use the Geometric Series test to determine convergence / divergence of a given geometric series
  • Compute the sum of a given (convergent) geometric series
  • Determine convergence / divergence of non-negative series by using the Cauchy Condensation Test, Comparison Test, Limit Comparison Test and Ratio Test
  • Study convergence / divergence of \(p\)-series
  • Prove that a given series converges absolutely
  • Prove that a complex series converges / diverges by using the Ratio Test for general series
  • Prove that a series converges conditionally
  • Use the Dirichlet / Alternate Convergence / Abel’s tests to study the convergence of a given series