Toronto Math Forum

MAT334--2020F => MAT334--Lectures & Home Assignments => Chapter 2 => Topic started by: Jessica Long on October 12, 2020, 02:35:52 PM

Title: Section 2.2 "closed form" Qs
Post by: Jessica Long on October 12, 2020, 02:35:52 PM

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. 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?

Title: Re: Section 2.2 "closed form" Qs
Post by: Victor Ivrii on October 13, 2020, 09:35:57 AM

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 $\frac{1}{(1-z)^m}$ ,...