Toronto Math Forum
MAT3342020F => MAT334Lectures & Home Assignments => Chapter 2 => Topic started by: Jessica Long on October 12, 2020, 02:35:52 PM

Questions 1418 ask us to find a "closed form" for each power series. I assume this is a nonpower series expression (e.g. e^{x}). Some of the power series seem to be variants on the geometric series, but then the closed form would only hold for some z based on the value of z, depending on the series. Would it be ok to just specify that the solution only holds for some z?

Yes, some of them are geometric series, and some of $e^{z}$, $\sin(z)$, $\sinh(z)$ and so on. However some can be derived from those, ether by substitution (f.e. $z^2$ instead of $z$), some by integration, differentiation, multiplication by $z^m$ or combination of both. F.e. consider geometric $\dfrac{1}{1z}$. Integratinfg we can get power series for $\Log (1z)$, diffeerentiating for $\frac{1}{(1z)^m}$ ,...