Lagrange remainder

$R_n(x)=f(x)-P_n(x)$ , bounded by $\dfrac{M}{(n+1)!}|x-a|^{n+1}$ — how far a Taylor polynomial can be from $f$ .

The Lagrange remainder $R_n(x)=f(x)-P_n(x)$ measures how far the degree- $n$ Taylor polynomial falls short of $f$ itself. Taylor's Theorem bounds it: if $|f^{(n+1)}(t)|\le M$ for all $t$ between $a$ and $x$ , then $|R_n(x)|\le\dfrac{M}{(n+1)!}|x-a|^{n+1}$ . Proving a Taylor series converges to $f$ (not just to some other value) means showing this remainder shrinks to $0$ as $n\to\infty$ .

Examples

For $\cos x$ centered at $0$ , using $|\cos^{(n+1)}(t)|\le1$ for all $t$ , the remainder satisfies $|R_n(x)|\le\dfrac{|x|^{n+1}}{(n+1)!}$ , which $\to0$ for every $x$ .