There's something hidden in it

Calculus Level 4

\[{a}_{n+1} +1 = (n+1){a}_{n} \]

Define an sequence \(\lbrace a_{n} \rbrace_{n=0}\) by the iteration above. For what value of \({a}_{0}\) will the iteration converge to some finite value?

Report your answer as first four digits of \({a}_{0}\) (without rounding off). For example, if your answer is \(3.141592653789 \ldots\) then enter 3.141.

Interestingly in the complete number line there exists only one possible value of \({a}_{0}\), for which this iteration doesn't diverge (that's the interesting aspect of it)

The sequence diverge to positive infinity for any starting value larger than \(a_0\), and the sequence diverge to negative for any starting value smaller than \(a_0 \).


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