Taylor Series Problem - Question and my attempt so far

[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?

 

[SammyS]

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

$\displaystyle f(x)=f(x_0)+f'(x_0)\Delta x<br /> +\frac{f''(x_0)}{2!}(\Delta x)^2+\frac{f'''(x_0)}{3!}(\Delta x)^3+\frac{f''''(x_0)}{4!}(\Delta x)^4+\dots\ ,\ \ \text{ where }\ \ \Delta x=x-x_0\ .$​