Question

A bit unclear about the factoring in this McClaurin Expansion, line 4 doesn't seem right to me

Answer 1

This is the reason I dropped AP Calc

Answer 2

that's an \(e^z\) Taylor expansion and the a multiple FOIL on the result, looks OK.

depending on what 5.24 actually says, they may have done one term too many on the expansion making it look worse than it needs be

negative big-O 's are funny

Answer 3

....but it might be neater to paste in the first few terms of the \(e^z\) expansions after doing this first:

\(= D(1 - 2 e^{- \beta x} + e^{- 2 \beta x})\)

Answer 4

it's still tedious, but maybe less so


\(= 1 - 2 \left(1 + (- \beta x) + \frac{(- \beta x)^2}{2} + \frac{(- \beta x)^3}{6} + \mathbb O (x^4) \right) ... \\ ... + \left(1 + (- 2\beta x) + \frac{(- 2 \beta x)^2}{2} + \frac{(- 2 \beta x)^3}{6} + \mathbb O (x^4) \right)\)

\(= 1 - 2 + 2 \beta x - (\beta x)^2 + \frac{( \beta x)^3}{3} + \mathbb O (x^4) + 1 - 2\beta x + 2 ( \beta x)^2 - \frac{4 }{3} (\beta x)^3 + \mathbb O (x^4) \)

\(= (\beta x)^2 - (\beta x)^3 + \mathbb O (x^4) \)