Discussion Overview
The discussion centers around the application of the absolute convergence test to the series $$\sum_{k=1}^{\infty}\left(\frac{k}{k+1}\right)^{2k}$$ and the related topic of finding the radius or interval of convergence for a function defined by a power series.
Discussion Character
- Exploratory
- Mathematical reasoning
- Homework-related
Main Points Raised
- One participant suggests computing the limit $$L=\lim_{k\to\infty}\left(\left(\frac{k}{k+1}\right)^{2k}\right)$$ to determine the behavior of the series, noting that if $$L\ne0$$, the series must diverge, but the test is inconclusive if the limit is zero.
- Multiple participants inquire about finding the radius or interval of convergence for the function $$f(x)=2k(x-1)^k$$, with one participant repeating the question and providing a computation for the radius of convergence as $$|x-1|<\lim_{k\to\infty}\left|\frac{2k}{2(k+1)}\right|=1$$.
- Another participant questions the treatment of the exponent $$k$$ in the context of the radius of convergence, implying a concern about its relevance.
- One participant responds by stating that only the coefficient of the exponential term is of concern and that constant factors in that coefficient can be disregarded.
Areas of Agreement / Disagreement
Participants express varying approaches to the absolute convergence test and the radius of convergence, with no consensus reached on the implications of the exponent or the absolute convergence test's application.
Contextual Notes
The discussion includes unresolved assumptions regarding the behavior of the series and the treatment of the exponent in the context of convergence, which may affect the conclusions drawn.
