# Taylor Series Problem - Question and my attempt so far

• jammyloller
In summary, the Taylor expansion of f(x) is a way to approximate the value of a function at a certain point by using information about the function's derivatives at that point. It involves calculating a series of terms, each representing a higher order of the function's derivative, and adding them together to get a more accurate approximation.
jammyloller
Question:

http://i.imgur.com/GsjeL.png

Here is my attempt so far:

http://i.imgur.com/AyOCm.png

Note: I've used m where the question has used j.

My attempt is based off some bad notes I took in class so the way I am trying to solve the problem may not be the best. I'm struggling to work out how to continue from the part where I've left a question mark, and I'm not even sure if the Taylor expansion is correct.

Could anybody offer some thoughts as to what to do?

jammyloller said:
Question:

http://i.imgur.com/GsjeL.png

Here is my attempt so far:

http://i.imgur.com/AyOCm.png

Note: I've used m where the question has used j.

My attempt is based off some bad notes I took in class so the way I am trying to solve the problem may not be the best. I'm struggling to work out how to continue from the part where I've left a question mark, and I'm not even sure if the Taylor expansion is correct.

Could anybody offer some thoughts as to what to do?
Hello jammyloller. Welcome to PF !

I'm not sure what your notes are supposed to say, but the Taylor expansion of f(x) about the point x0 is given by
$\displaystyle f(x)=f(x_0)+f'(x_0)\Delta x +\frac{f''(x_0)}{2!}(\Delta x)^2+\frac{f'''(x_0)}{3!}(\Delta x)^3+\frac{f''''(x_0)}{4!}(\Delta x)^4+\dots\ ,\ \ \text{ where }\ \ \Delta x=x-x_0\ .$​

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## What is a Taylor Series?

A Taylor Series is a mathematical representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point.

## What is the purpose of finding a Taylor Series?

The purpose of finding a Taylor Series is to approximate a function using a polynomial, which can make it easier to evaluate or manipulate the function for certain calculations.

## How do you calculate a Taylor Series?

To calculate a Taylor Series, you need to know the function's derivatives at a single point, which is usually chosen to be 0. Then, you can use the formula for the Taylor Series to find the coefficients of the polynomial.

## What is the formula for a Taylor Series?

The formula for a Taylor Series is:
f(x) = f(a) + f'(a)(x-a) + f''(a)(x-a)^2/2! + f'''(a)(x-a)^3/3! + ... + fn(a)(x-a)^n/n! + ...

## How accurate is a Taylor Series approximation?

The accuracy of a Taylor Series approximation depends on the number of terms used in the series. The more terms included, the more accurate the approximation will be. However, the Taylor Series will only be an exact representation of the original function if an infinite number of terms are used.

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