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

[PeterP]

Challenge 2, HP41, Sandbox module, unsophisticated, straight forward RPN.

Lbl 'AS
2021
STO 00 ;HP 41 is very slow with numbers
2
+
1/x
STO 01 ; First factor,
STO 02 ; Sum
2
St* 01 ;build second factor
RCL 00
3
STO 11 ;Current Prime
+
ST/ 01
GTO 05 ;calculate sum
LBL 06 ;find next prime
RCL 11 ; current prime
STO L
LBL 00 ;prime finding loop
LastX
2
+
PRIME?
GTO 01
GTO 00
LBL 01
View X ;show current prime
X <> 11 ; Store current prime
X <> 10 ; Put old current prime into last Prime
LBL 02
RCL 10 ; Last Prime
ST* 01
RCL 00
RCL 11
+
ST/ 01 ; update current factor
LBL 05 ; calc sum
RCL 01
ST+ 02
View 02 ; View Current Sum
GTO 06

The sum very quickly converges to 1/2021 or, to be more precise, after some sleuthing using different constants, to 1/A with A being the constant added in the denominator. Some sleuthing shows that its also irrelevant that the sum uses the prime numbers, one could use any sequence of numbers, even a constant 1 (ie sum of 1/(2021+1)^k).

doing a little bit more paper sleuthing on this, one can see that the numerator becomes a O(A^n) and the denominator O(A^(n+1)), meaning the sum trends to 1/A.