IB Mathematics Analysis and Approaches 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 $\mathrm{artanh}(x)$ 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 $\arctan \left(\frac{x}{2}\right)$ 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 $\ln \left(\frac{1+x}{1-x}\right)$.