§8.5 Finding and using Taylor series (continued)

Summary

If $e^x = a_0+a_1x+a_2x^2+\cdots$, differentiating term by term and comparing with the original shows

The Fresnel integral $C(x) = \int_0^x\cos(t^2)\,dt$ has series built from $\cos(t) = 1-\frac{t^2}{2!}+\frac{t^4}{4!}-\cdots$ by substituting $t^2$ and integrating, giving

To evaluate a $\frac{0}{0}$ limit such as $\lim_{x\to0}\frac{\sin(x)-x}{x^3}$, the Taylor method says to

Using $C(x) = x-\frac{x^5}{5\cdot2!}+\frac{x^9}{9\cdot4!}-\cdots$ through the $x^5$ term, estimate $C(0.5) = \frac{159}{320}$ as a decimal, rounded to four decimals.

Using the expansion $\sin(x) = x-\frac{x^3}{3!}+\cdots$, compute $\lim_{x\to0}\frac{\sin(x)-x}{x^3}$ as a decimal, rounded to three places.

Using the expansion $e^{x^2} = 1+x^2+\frac{x^4}{2!}+\cdots$, compute $\lim_{x\to0}\frac{e^{x^2}-1}{x^2}$.

Differentiating the series $1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots$ term by term reproduces exactly the same series.