# Riemann Integration and Series of Functions

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## Riemann Integration and Series of Functions at Swayam Overview

Duration | 15 weeks |

Start from | Start Now |

Mode of learning | Online |

Difficulty level | Beginner |

Official Website | Go to Website |

Credential | Certificate |

## Riemann Integration and Series of Functions at Swayam Highlights

- Earn a certification after completion
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## Riemann Integration and Series of Functions at Swayam Course details

- Those pursuing advanced degrees in mathematics or applied mathematics, who need a deeper understanding of integration techniques and the convergence of series of functions as part of their advanced analysis coursework

- Students will be able to calculate the Riemann integral of a function over a closed interval using the definition and standard techniques
- Students will be able to prove basic properties of the Riemann integral, such as linearity, additivity, and monotonicity
- Students will be able to define pointwise and uniform convergence of series of functions and explain the differences between them
- Students will be able to apply various tests and criteria to determine the convergence of series of functions, including the Weierstrass M-test and the Cauchy criterion for uniform convergence

- The course "Riemann Integration and Series of Functions" is proposed for B.Sc Mathematics or B.Sc. (Hons) Mathematics students. The course content is divided in to 39 modules and the course credit is four
- The first part of the course discusses Riemann's theory of integration. It starts with the definition of the Riemann sum, which naturally leads to the notion of integrals, discusses equivalent conditions for the existence of integral and properties of integral and finally proves the 'Fundamental Theorem of Calculus'
- The second part of the course is on the sequence and series of functions, where we will look at the significance of 'uniform convergence' to prove the continuity, differentiability and integrability of the limit function of a sequence of functions. Finally, we will define limit superior and limit inferior and discuss results for the special case of 'power series'

## Riemann Integration and Series of Functions at Swayam Curriculum

**Week 1**

Module 1: Introduction to Riemann integration, Darboux sums.

Module 2: Inequalities for upper and lower Darboux sums.

Module 3: Darboux integral

Interaction based on the three modules covered

Subjective Assignment

**Week 2**

Module 4: Cauchy criterion for integrability

Module 5: Riemann’s definition of integrability

Module 6: Equivalence of definitions.

Interaction based on the three modules covered

Subjective Assignment

**Week 3**

Module 7: Riemann integral as a sequential limit

Module 8: Riemann integrability of monotone functions and continuous functions

Module 9: Further examples of Riemann integral of functions

Interaction based on the three modules covered

Subjective Assignment

**Week 4**

Module 10: Algebraic properties of Riemann integral

Module 11: Monotonicity and additivity properties of Riemann integral

Module 12: Approximation by step functions

Interaction based on the three modules covered

Subjective Assignment

**Week 5**

Module 13: Mean value theorem for integrals

Module 14: Fundamental Theorem of Calculus (first form)

Module 15: Fundamental Theorem of Calculus (second form)

Interaction based on the three modules covered

Subjective Assignment

**Week 6**

Module 16: Improper integrals of Type-1.

Module 17: Improper integrals of Type-2 and mixed type.

Module 18: Gamma and beta functions

Interaction based on the three modules covered

Subjective Assignment

**Week 7**

Module 19: Pointwise convergence of a sequence of functions

Module 20: Uniform convergence

Module 21: Uniform norm

Interaction based on the three modules covered

Subjective Assignment

**Week 8**

Module 22: Cauchy criterion for uniform convergence

Module 23: Uniform converegnce and continuity

Module 24: Uniform convergence and integration

Interaction based on the three modules covered

Subjective Assignment

## Riemann Integration and Series of Functions at Swayam Faculty details

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