Revision sheet

Taylor polynomials

$P_n(x) = \displaystyle\sum_{k=0}^{n} \dfrac{f^{(k)}(a)}{k!}(x - a)^k$.

Watch out: divide by $k!$; use powers of $(x - a)$.

Lagrange error bound

$|R_n| \le \dfrac{M|x - a|^{n+1}}{(n + 1)!}$, $M \ge |f^{(n+1)}|$.

Watch out: $n + 1$ everywhere.

Radius and interval

1. Ratio test → $|x - a| < R$.
2. Test $x = a - R$ and $x = a + R$ separately.

Watch out: solve for $|x - a|$ carefully; never skip the endpoints.

Known Maclaurin series

$e^x$, $\sin x$ (odd), $\cos x$ (even), $\dfrac{1}{1 - x}$.

Watch out: signs alternate in sin and cos.

New series from old

• Multiply by $x^k$: shift the powers.
• Substitute: replace $x$ by $u(x)$.
• Differentiate or integrate term by term.

Watch out: integrating divides by the new power.