Hi all --
- 1 -----
Inspired by lately reading up on the Kempner series I tried this
modification (depletion) of the harmonic series:
fib >: i.13
1 1 2 3 5 8 13 21 34 55 89 144 233
% fib >: i.13
1 1 1r2 1r3 1r5 1r8 1r13 1r21 1r34 1r55 1r89 1r144 1r233
+/ % fib >: i.13
851897554247r254074700880
^^. +/ % fib >: i.13
3.35294128575735
NB. ...
^^. +/ % fib >: i.50
3.3598856661144394
^^. +/ % fib >: i.60
3.3598856662422096
^^. +/ % fib >: i.70
3.3598856662429739
^^. +/ % fib >: i.80
3.3598856662433558
and have been wondering about convergence/divergence.
- 2 -----
In parallel I did this
fib i.13
0 1 1 2 3 5 8 13 21 34 55 89 144
ecf fib i.13
2882971364492r4895735924493
^^. ecf fib i.13
0.588874
NB. (fib) producing Fibonacci numbers
NB. (ecf) evaluating a Continued Fraction
and have wondered whether this constant had a name (since the Fib
numbers themselves are fairly famous), and in what other contexts it
might pop up.
NB. The Wolfram|Alpha equivalent would have been
NB. FromContinuedFraction[Fibonacci[Range[0,13]]]
-----
Could you shed some light on these (while keeeping in mind that I'm
not a mathematician).
Thanks (and with season's greetings)
-M
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