Given $a_1=\frac{1}{3},a_{n+1}=a_n+\frac{a_n^2}{n^2}$，work out $\displaystyle\lim_{n \to \infty}{a_n}$（I think it's $\frac{6}{\pi^2}$,but I can't prove it)
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Your conjecture seems to be incorrect. $a_{1000}\approx0.609476>\frac6{\pi^2}\approx0.607927$. The actual limit appears to be closer to $0.609848$. – user170231 Nov 04 '22 at 16:10
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It's possible to prove that the limit exists. But I don't know whether the limit can be computed. How do we know that the limit can be computed analytically? – NN2 Nov 04 '22 at 16:33
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Does this answer your question? Prove the existence of limit of $x_{n+1}=x_n+\frac{x_n^2}{n^2}$ – zwim Nov 04 '22 at 16:52