- #1

- 12

- 0

## Homework Statement

I have a problem with McLaurin series. I never know when to stop. How do I know if O(x

^{3}) is adequate, or O(x

^{5})?

Let's take this exam question as an example.

[tex]

\lim_{x \to 0} \frac{(x+1)e^x -1-2x}{cosx-1}

[/tex]

[tex]

\frac{(x+1)e^x -1-2x}{cosx-1} = \frac{(x+1)(1+x+\frac{x^2}{2!}+...) -1-2x}{1-\frac{x^2}{2!}+...-1} =

[/tex]

[tex]

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

[/tex]

[tex]

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

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

[/tex]

How do I know to stop at O(x

^{3}) for e

^{x}and at O(x

^{4}) for cosx? Any tactics?

## Homework Equations

[tex]

e^x = 1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+...+ \frac{x^n}{n!}+O(x^{n+1})

[/tex]

[tex]

cosx = 1+x+\frac{x^2}{2!}+\frac{x^4}{4!}+...+(-1)^{n-1} \frac{x^{2n}}{2n!}+O(x^{2n+2})

[/tex]

## The Attempt at a Solution

...?

Last edited: