MTH632 Quiz Modules 59-98

LECTURE 30 · MODULE 59 · PART I

The purpose of discussing the power series in this course is to construct the complex valued functions with the help of power series. The power series also help to explore new structure of the complex valued functions.

Before constructing the complex valued function from the infinite series, let us look at what a function is.

What is a function?

A function is just like a machine in which we give input and then we get output. We give in a complex number z and obtain a complex number f(z). How can a series be used to define a function?

First, we look at what an infinite series is.

An infinite series an expression of the form

𝑛=1

Consider an infinite series of the form

𝑛=0

WARNING

The series

𝑛=0 defines a function only if it converges.

MCQ · MCQ · MCQ · POWER SERIES FUNCTION

The power series function is given by

𝑛=0 For properly defining a function with the help of power series, we need to define its domain. Since a function on an infinite series can be made if the infinite series is convergent. If an infinite series is divergent, then it is useless to define a function on it.

Therefore, the Domain of the power series function must consists of the collection of all points 𝑧 for which the series converges. agar infinite series convergent hy to function define ho ga agar series divergent hy to function define karna useless hy