Analysis

Learning Outcomes

Learning Outcomes

Having successfully completed this module you will be able to:

• Prove using the definition that a given sequence converges to a given limit
• Determine whether a given improper integral converges
• Determine whether a series of positive terms converges, either by explicitly summing the series or by using a test, such as the comparison test, the ratio test, the root test, or the integral test
• Define an improper integral
• Determine a radius and interval of convergence for a given power series
• Understand when one can differentiate and integrate a power series
• Find the Taylor series of a given function
• Determine whether a sequence of real numbers converges, either by evaluating the limit directly or by showing the sequence is bounded and monotone

• Real numbers, bounded subsets of the reals, supremum and infimum, the Archimedean property of the real numbers.
• Sequences and their limits, arithmetic of limits, inequalities for limits.
• Bounded monotonic sequences, convergence and divergence criteria for sequences.
• The Bolzano-Weierstrass theorem. Completeness of the set of the reals.
• Continuity and uniform continuity of functions.
• Sequence and series of functions.
• Convergence of series by partial sums, geometric and harmonic series, algebra of series.
• Series with non-negative terms: comparison, integral, ratio, root tests.
• Absolute and conditional convergence, alternating test.
• Real power series, radius and interval of convergence; uniqueness, algebra, differentiation, integration of power series.
• Uniform convergence of sequences and series of continuous functions. The Weierstrass M-test. (if time)
• Cauchy mean value theorem, and convergence of Taylor series.
• Riemann Integral and the Fundamental Theorem of Calculus.
• Improper integrals.

Lectures, problem classes, workshops, private study

Summative

This is how we’ll formally assess what you have learned in this module.

Referral

This is how we’ll assess you if you don’t meet the criteria to pass this module.

Repeat

An internal repeat is where you take all of your modules again, including any you passed. An external repeat is where you only re-take the modules you failed.