Taylor Series:

Find the Taylor series of the following functions at the given $b$:

• $f(x) = \ln | 1 + x^2 |$ at $b = 0$
• $f(x) = \ln \left( \frac{1+x}{1-x} \right)$ at $b = 0$
• $f(x) = \frac{\sin^2(x)}{3x}$ at $b = 0$
• $f(x) = \frac{x^6}{(1-x^2)^3}$

Limits:

Compute the following limit: $$ \lim_{x \rightarrow 0} \frac{e^x -1 -x}{x^2}.$$

Estimating Integrals:

Evaluate $\int_0^1 e^{-x^2} dx$ to within an error of $0.001$.