Discussion Overview
The discussion revolves around determining the convergence or divergence of the series \(\sum \frac{2}{\sqrt{n}+1}\) and a related series \(\sum \frac{1}{\sqrt{n}+\sqrt{n+1}}\). Participants explore various tests for convergence and share their reasoning and challenges in applying these tests.
Discussion Character
- Debate/contested
- Mathematical reasoning
- Homework-related
Main Points Raised
- One participant questions the convergence of the series \(\sum \frac{2}{\sqrt{n}+1}\) and mentions that the divergence test yields a result of 0, leading to confusion about which test to apply.
- Another participant suggests comparing the series to \(\sum \frac{1}{n}\) as a potential method for analysis.
- A participant expresses uncertainty about the correct answer being convergent, prompting a clarification on the limits of the series.
- A later reply indicates that the limit comparison test with \(\sum \frac{1}{n}\) was successful in determining divergence.
- One participant acknowledges the divergence of the harmonic series \(\sum \frac{1}{n}\) and corrects their earlier assertion about the series being convergent.
- The discussion shifts to a new series \(\sum \frac{1}{\sqrt{n}+\sqrt{n+1}}\), with a participant expressing difficulty in finding a suitable comparison series.
- Another participant suggests considering the behavior of the terms as \(n\) becomes large to identify an appropriate comparison.
Areas of Agreement / Disagreement
Participants do not reach a consensus on the convergence of the initial series, with some asserting divergence and others initially suggesting convergence. The discussion on the second series remains unresolved, with participants exploring potential comparison series without agreement on a conclusion.
Contextual Notes
Participants express uncertainty regarding the application of various convergence tests and the choice of comparison series, indicating that assumptions about limits and series behavior are not fully established.