Table of Contents
Explain in Detail :Proof of Taylor’s Series Theorem
We will use the definition of a Taylor series to show that the series is convergent.
A Taylor series is a series of the form:
where a is a real number, x is a real number, and n is a positive integer.
The series is convergent if the limit of the sequence of partial sums is finite.
We will use the following fact:
The limit of the sequence of partial sums is finite if and only if the series is convergent.
We will show that the series is convergent.
We will first show that the limit of the sequence of partial sums is finite.
We will do this by using the Squeeze Theorem.
The Squeeze Theorem states that if f is continuous on the interval [a,b] and g is continuous on the interval [c,d], then the following inequality holds:
We will use the Squeeze Theorem to show that the limit of the sequence of partial sums is finite.
We will start by assuming that the limit of the sequence of partial sums is not finite.
This will allow us to find a number M such that for all n greater than or equal to M, the difference between the partial sum and the limit of the sequence of partial sums is greater than or equal to 0.
We will then use the fact that the series is convergent to find a number N such
Taylor Series
The Taylor series is a powerful tool for approximating the value of a function at a given point.
The Taylor series is a representation of a function as a sum of terms, each of which is a product of the function and a power of x.
The Taylor series for a function can be used to approximate the value of the function at a given point.
The order of the Taylor series is the degree of the power of x in the terms of the series.
The Taylor series for a function can be used to approximate the value of the function at a given point to any degree of accuracy.
