Struggling with a Taylor Series problem!

[jammyloller]

Question:

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Here is my attempt so far:

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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?

 

[HallsofIvy]

What you show isn't a "Taylor series" at all or even a power series. You have everything, including the variable, inside the derivative functions.

The Taylor series for f(x), about \(\displaystyle x_0\), is \(\displaystyle \sum_{n=0}^\infty \frac{f^{(n)}(x_0)}{n!}(x- x_0)^n\). You should have the derivatives of f evaluated at \(\displaystyle x_0\) and then \(\displaystyle (x- x_0)= ((x_0+ \Delta x)- x_0)^n= (\Delta x)^n\).

And you don't want to "substitute the expansion" into the given formula- you want to derive the formula from the expansion.