Question Type 2: Maclaurin series by differentiating or integrating known series Exercises

Find the Maclaurin series for $f(x)=\frac{1}{4+x^2}$ by differentiating the Maclaurin series for $\arctan\left(\frac{x}{2}\right)$.

Find the Maclaurin series for $\sqrt{1+x}$ using the binomial theorem.

Find the Maclaurin series for $f(x)=(1-x)^{-2}$ by differentiating the series for $\frac{1}{1-x}$.

Obtain the Maclaurin series for $\operatorname{artanh}(x)$ by integrating the series for $\frac{1}{1-x^2}$.

Obtain the Maclaurin series for $\ln(1+x)$ by integrating the series for $\frac{1}{1+x}$.

Derive the Maclaurin series for $\arctan\bigl(\tfrac{x}{2}\bigr)$ using the known series for $\arctan(x)$.

Find the Maclaurin series for $f(x) = \frac{1}{1 + x^2}$.

Derive the Maclaurin series for $\arcsin(x)$ by integrating the series for $(1-x^2)^{-\frac{1}{2}}$.

Using the binomial series for $(1 - x)^{-\frac{1}{2}}$, differentiate it to obtain the Maclaurin series for $(1 - x)^{-\frac{3}{2}}$.

Find the Maclaurin series for $\ln\left(\tfrac{1+x}{1-x}\right)$.