As for Q2:
Thanks to Raul, who reminded me that rationals are part of the
extended precision environment.
I found this entry OEIS A073822 (http://oeis.org/A073822) dated 2002-Aug-12 :
"Decimal expansion of number with continued fraction expansion 0,
1, 1, 2, 3, 5, ... (the Fibonacci numbers)"
5, 8, 8, 8, 7, 3, 9, 5, ...
The cited paper by Marek Wolf "Continued fractions constructed from
prime numbers" states on p. 20:
"The continued fraction build from Fibonacci numbers an = Fn
F = [0; 1; 1; 2; 3; 5; 8; ... ; Fn; ... ] =
0:588873952548933507671231121246787384 ...
appears at the Sloane The On-Line Encyclopedia of Integer Sequences
as the entry A073822. Apparently ... F also should be
transcendental, but we are not aware of the proof of this fact."
Searching for "0.5888739525" or "0 5 8 8 8 7 3 9 5 2 5" had not
returned any results -- but something like "8 8 8 7 3 9 5 2 5" finally did.
Key to finding it was this hint from the OEIS Wiki: "Look up the
sequence, omitting the first couple of terms. Look up terms 2 through 12, say."
I have been pondering over this for quite a while (some months); glad
I found some answers. Case closed, sort of.
-M
At 2022-12-29 10:17, you wrote:
As for Q1:
No luck in searching for something like 3.3598856662 .
What finally produced some relavant entries was
(a) using search words 'fibonacci' and 'reciprocal', or
(b) writing the number explicitely as "3 3 5 9 8 8 5 6 6 6 2" oeis .
This led me to e.g. OEIS A079586 (https://oeis.org/A079586) .
Means that the series *converges* to what Eric Weisstein calls the
'Reciprocal Fibonacci Constant'
(https://mathworld.wolfram.com/ReciprocalFibonacciConstant.html) .
As for Q2:
No result so far; will report back.
Thanks.
-M
At 2022-12-26 18:39, you wrote:
I don't know about naming here, but if you are going to attempt
searching on numeric results, you might want something more precise
than ^^. to obtain a decimal representation.
For example:
0j16 ": 851897554247r254074700880
3.3529412857573449
I hope this helps,
--
Raul
On Mon, Dec 26, 2022 at 6:50 AM Martin Kreuzer <[email protected]> wrote:
> 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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