[VA] Short & Sweet Math Challenge #25 "San Valentin's Special: Weird Math"

[Nihotte(lma)]

(02-19-2021 11:32 PM)Valentin Albillo Wrote:        
Hi, all:

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Concoction the First: Weird limit

      Point d is still unaddressed:  d. Can you explain the constant component of said [asymptotic] expression ?
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As stated, Last Chance. Thanks and best regards.  
V.

I completed my initial result by calculating the rates of increase between the different successive results obtained.
If I consider each f(i) and if I calculate the Rate for i by (f(i) - f(i-1)) / (i - i-1), it will be :
f(1) = 2.71959
f(2) = 4.67827 ---> 1.95868 / (2 - 1) = 1.95868
f(2.021) = 4.71806 ---> 0.03979 / (2.021 - 2) = 1.89476
f(3) = 6.66808 ---> 1.95002 / (3 - 2.021) = 1.99185
f(pi) = 6.95027 ---> 0.28219 / (pi - 3) = 1.99297
f(4) = 8.66601 ---> 1.71574 / (4 - pi) = 1.99843
f(5) = 10.66641 ---> 2.0004 / (5 - 4) = 2.0004
f(5+1/6) = 10.99947 ---> 0.33306 / (5+1/6 - 5) = 1.99836
f(10) = 20.65914 ---> 9.65967 / (10 - (5+1/6)) = 1.99855
f(15) = 30.667 ---> 10.00786 / (15 - 10) = 2.001572
f(20) = 40.66927 ---> 10.00227 / (20 - 15) = 2.000454
and I can see that each time the ratio gives a result really near 2
Each time ? No !
There is one exception with (f(2.021) - f(2)) / (2.021 - 2) = 1.89476
Even, if you evaluate (f(3) - f(2))/ (3 - 2) = 1.98981, the result keeps a sign of this particularity
I can't say if it would possible to notice this lower result in other places that are independent or distant from 2.021 (excluding the portion before i=3, of course), but I can say that is already weird !