Taylor expansions at 0: the 6 essentials for prep classes

The 6 fundamental expansions at 0 (order 3)

$e^x = 1 + x + \dfrac{x^2}{2} + \dfrac{x^3}{6} + o(x^3)$

$\ln(1 + x) = x - \dfrac{x^2}{2} + \dfrac{x^3}{3} + o(x^3)$

$\sin(x) = x - \dfrac{x^3}{6} + o(x^3)$

$\cos(x) = 1 - \dfrac{x^2}{2} + o(x^3)$

$(1 + x)^\alpha = 1 + \alpha x + \dfrac{\alpha(\alpha - 1)}{2} x^2 + o(x^2)$

$\dfrac{1}{1 - x} = 1 + x + x^2 + x^3 + o(x^3)$

What Taylor expansions are for

They let you replace a function by a polynomial near 0, making $\dfrac{0}{0}$-type limits trivial.

A satisfying example

$\displaystyle \lim_{x \to 0} \dfrac{e^x - 1 - x}{x^2}$

Numerator: $e^x - 1 - x = \dfrac{x^2}{2} + o(x^2)$.

So the limit is $\displaystyle \lim_{x \to 0} \dfrac{x^2/2}{x^2} = \dfrac{1}{2}$.

The 3 golden rules

1. Expand to a sufficient order — otherwise you lose the key info.
2. Never add $o(x^n)$ of different orders without thinking.
3. Check that your expansion is really at 0 (else make a change of variable).

The composition trick

For $\ln(1 + \sin x)$, substitute $\sin x$ for $x$ in $\ln(1+u)$, then substitute the $\sin$ expansion and regroup by order.

?Your turn

What is the Taylor expansion of sin(x) − x to order 3 at 0?

A.−x²/2 + o(x³)B.−x³/6 + o(x³)C.0 + o(x³)D.x³/6 + o(x³)

Check