MAT334--2020F > Chapter 2

Section 2.2 "closed form" Qs


Jessica Long:
Questions 14-18 ask us to find a "closed form" for each power series. I assume this is a non-power series expression (e.g. ex). 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?

Victor Ivrii:
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 $\dfrac{1}{1-z}$. Integratinfg we can get power series for $-\Log (1-z)$, diffeerentiating for $\frac{1}{(1-z)^m}$ ,...


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