The discussion centers on the convergence of the sequence defined by \( a_{n+1} = a_n - a_n^2 \) with \( a_0 \in (0,1) \). Participants question whether \( \lim_{n\rightarrow\infty} n a_n \) exists and seek various solutions. A key point raised is the importance of demonstrating one's own solution before requesting assistance, emphasizing the collaborative nature of homework help forums.
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grusini said:Homework Statement
Let [itex](a_n)_{n\in\mathbb{N}}[/itex] be a real sequence such that [itex]a_0\in(0,1)[/itex] and [itex]a_{n+1}=a_n-a_n^2[/itex]
Does [itex]\lim_{n\rightarrow\infty}na_n[/itex] exist? If yes, calculate it.
Homework Equations
The Attempt at a Solution
I have a solution but I'd like to see other solutions..