Finding and using Taylor series

Use known Taylor series and substitution and algebraic techniques to find a Taylor series representation for each of the following functions. In addition, state the interval of $x$-values for which you expect each Taylor series to converge.

1. $g(x) = x^3 \sin(x^2)$

2. $h(x) = e^{-x^4}$

3. $p(x) = \dfrac{1}{1+5x}$

4. $q(x) = x^2\ln(1+x^4)$

5. $r(x) = \dfrac{e^{3x}-1}{3x}$

In this activity we will determine the Taylor series expansion for the famous error function, , that arises in probability theory and other important applications.

1. Use the Taylor series for $e^{x}$ to find the Taylor series for $e^{-t^2}$.

2. Next, evaluate the integral $\int_0^x e^{-t^2} \, dt$ by replacing $e^{-t^2}$ with its Taylor series.

3. Use your work in (b) to state the Taylor series for $\erf(x) = \frac{2}{\sqrt{\pi}} \int_0^x e^{-t^2} \, dt$

4. For what interval of $x$-values will the Taylor series for…

In this exercise we find the Taylor series representation for two famous functions, the Fresnel integral functions and . The Fresnel integral functions are important in optics and are used in the design of Fresnel lenses such as those found in lighthouses along the Lake Michigan shore.

1. Use the Taylor series for $\cos(x)$ to find the Taylor series for $\cos(t^2)$ and hence write $C(x)$ as a Taylor series.

2. For what interva…