Facing problem in analysing Taylor series expansion

In summary, the conversation discusses the process of finding the Taylor series expansion of a function. It is mentioned that when finding the expansion of a function with a variable inside, such as sin(x^2), the variable is simply replaced in the expansion of the original function. The conversation also addresses the question of why it should not come out differently and suggests that the terms may look different, but should ultimately be able to be put in the same form.
  • #1
bibhatsucool
1
0
This is a very basic question .
Actually in Taylor series expansion of say "sin x" we write the expansion ... (as it is,I am not writing it)
But when we are asked to write the expansion of sin(x^2) we just replace 'x' by "x^2" in the expansion of sin x.
Or if asked some other function such as Sin(f(x)) ,we just replace x by f(x) in Taylor expansion of Sin(x) ...
Instead of finding the expansion of g(x) =Sin(f(x)) term by term mechanically ?why ?
also I found it come out to be different... !
Help guys
 
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  • #2
It should not come out differently, if it does you are not doing it correctly. Note that the terms may look different in different sums, but essentially you should always be able to put it on the same form.
 

Related to Facing problem in analysing Taylor series expansion

What is a Taylor series expansion?

A Taylor series expansion is a way to represent a mathematical function as an infinite sum of terms. It is named after the mathematician Brook Taylor and is a useful tool in approximating functions and solving problems in calculus.

Why might someone face problems in analysing a Taylor series expansion?

There can be a variety of reasons why someone may face difficulties in analysing a Taylor series expansion. Some common challenges include understanding the concept of a Taylor series, determining the appropriate number of terms to use, and dealing with convergence issues.

How can I determine the appropriate number of terms to use in a Taylor series expansion?

The number of terms used in a Taylor series expansion depends on the desired level of accuracy. Generally, the more terms used, the more accurate the approximation will be. However, it is important to consider the convergence of the series and whether adding more terms will significantly improve the accuracy.

What are some common convergence tests used in analysing a Taylor series expansion?

There are several convergence tests that can be used to determine if a Taylor series expansion is convergent or not. These include the Ratio Test, Root Test, and the Integral Test. It is important to understand and apply these tests correctly in order to analyse the convergence of a Taylor series.

Are there any alternative methods to analysing Taylor series expansions?

Yes, there are alternative methods to analysing Taylor series expansions such as using Maclaurin series (a special case of Taylor series), power series, and Fourier series. These methods may be more suitable for certain functions or problems, so it is important to understand and be familiar with all of them.

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