AP Calculus BC / Unit 10: Infinite Sequences and Series
10.15 Representing Functions as Power Series
Notes
Build new series from known ones:
- Substitute: $\displaystyle e^{x^2}=\sum\frac{x^{2n}}{n!}$.
- Multiply: $\displaystyle x\sin x=x^2-\frac{x^4}{3!}+\dots$
- Differentiate or integrate term by term (the radius stays the same).
Practice
The Maclaurin series for $e^{-x^2}$ starts:
Substitute $-x^2$ into $e^u$.
Integrating $\displaystyle \frac{1}{1+x^2}=1-x^2+x^4-\dots$ term by term gives the series for:
$\displaystyle \int\frac{dx}{1+x^2}=\arctan x$.
Differentiating the series for $\sin x$ gives the series for:
Term by term derivative matches $\cos x$.
Ready for more? Take the Unit 10 test.