[seqfan] Can π be approximated in this manner?

[Alonso Del Arte]

Obviously 1 + 1 = 2. Then 2 + 1/2 = 5/2, 5/2 + 2/5 = 29/10, 29/10 + 10/29 =
941/290, etc. This leads to A073833 / A073834. I don't know if that
converges to anything in particular or just keeps growing. But the thought
occurs to me that if sequences of this type do converge, maybe there's one
that approximates π, or π multiplied by some obvious positive rational
number.

I tried a few, without much success, e.g.,

scala> res27.take(10).toList
res28: List[fractions.BigFraction] = List(1/30, 901/30, 812701/27030,
661213536301/21967308030, 437685903209758747243501/14525081425529653797030,
191779927858960569002522847057449146662223557901/6357423382928236658859400112096701350640602030,
...

scala> res28.map(_.getDouble)
res29: List[Double] = List(0.03333333333333333, 30.033333333333335,
30.06662967073622, 30.099889135164094, 30.133111848892693,
30.166297933523268, 30.199447509987827, 30.23256069855425,
30.265637618831395, 30.29867838977414)

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Alonso del Arte
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