Discussion Overview
The discussion revolves around determining whether the statement \(2^n = O(n!)\) is true or false, with participants exploring how to prove this relationship. The focus is on mathematical reasoning and the application of big O notation.
Discussion Character
- Mathematical reasoning
- Homework-related
- Exploratory
Main Points Raised
- One participant suggests that \(2^n = O(n!)\) might be true based on their understanding of growing order functions but admits they cannot prove it.
- Another participant proposes showing that \(n! > 2^n\) for all \(n\) above a certain value as a way to prove the statement.
- A participant expresses difficulty in establishing the link between factorial and exponential functions, specifically questioning how to choose values for \(k\) and \(n_0\) in the big O definition.
- Further hints are provided about examining specific values of \(n\) to identify a pattern, indicating that \(2^n\) is greater than \(n!\) for small \(n\) but less for larger \(n\).
- Another participant emphasizes that it is sufficient to find a specific \(k\) (suggesting \(k=1\)) to prove the inequality holds for sufficiently large \(n\).
Areas of Agreement / Disagreement
Participants do not reach a consensus on the proof of the statement. There are multiple viewpoints on how to approach the proof, and uncertainty remains regarding the specific values of \(k\) and \(n_0\) needed.
Contextual Notes
Participants express limitations in their understanding of the relationship between factorial and exponential growth, and there are unresolved questions about the choice of constants in the big O notation.