Questions

01

Power series calculus

Use calculus and the first familiar example $(1-x{)}^{-1}$ to find the power series that equals each of:

• $(1+x{)}^{-1}$
• $\ln (x-1)$
• $\ln (x+1)$
• $\arctan (x)$

02

Bessel function

Show that the series

$$y(x)=1-\frac{x^2}{2^2}+\frac{x^4}{2^2{4}^2}-\frac{x^6}{2^2{4}^2{6}^2}+\cdots .$$

satisfies the ODE $x{y}^{\prime \prime }+y^{\prime }+xy=0$.

03

Method of power series

Find a power series solution to $y^{\prime }=2xy$. Then solve the equation algebraically using prior methods we have learned, and compare your answers.

04

Method of power series

Find a power series solution to $y^{\prime }-y=x^2$. Then solve the equation algebraically using prior methods we have learned, and compare your answers.