Discussion Overview
The discussion revolves around the convergence of the series $$\sum_{n=2}^{\infty} \frac{a_{n}}{n^{2/3}}$$ given that $$\sum_{n=1}^{\infty} a^2_{n}$$ converges and that $$a_{n}$$ is non-negative. Participants explore whether the statement is true or false, and they discuss various approaches to prove or disprove it, including the use of the Cauchy-Schwarz inequality and convergence tests.
Discussion Character
- Exploratory
- Technical explanation
- Debate/contested
- Mathematical reasoning
Main Points Raised
- Some participants suggest that if $$\sum_{n=1}^{\infty} a^2_{n}$$ converges, then the series $$\sum_{n=2}^{\infty} \frac{a_{n}}{n^{2/3}}$$ should also converge, proposing that the terms of the latter series are dominated by the former.
- Others argue that specific choices of $$a_n$$, such as $$a_n^2 = \frac{1}{n^{7/6}}$$, could lead to different conclusions about the convergence of $$\sum_{n=2}^{\infty} \frac{a_{n}}{n^{2/3}}$$.
- A participant mentions the Cauchy-Schwarz inequality as a potential method to prove the convergence of the series, suggesting that it can be applied to relate the two series.
- Another participant expresses uncertainty about using the Cauchy-Schwarz inequality, indicating that they have only learned other convergence tests and feel unprepared to apply this method.
- One participant provides a detailed application of the Cauchy-Schwarz inequality to show how it can lead to a conclusion about the convergence of the series.
- There is a mention of the Direct Comparison Test in conjunction with the Cauchy-Schwarz inequality as a way to establish convergence.
Areas of Agreement / Disagreement
Participants do not reach a consensus on the convergence of the series $$\sum_{n=2}^{\infty} \frac{a_{n}}{n^{2/3}}$$. While some propose methods to prove convergence, others raise counterexamples and express uncertainty about the applicability of certain techniques.
Contextual Notes
Some participants note limitations in their understanding of the Cauchy-Schwarz inequality and other convergence tests, which may affect their ability to engage fully with the problem.