Saturday, February 3, 2024

Swiss Micros DM42 and TI-84 Plus: Sum of an Infinite Geometric Series

Swiss Micros DM42 and TI-84 Plus:  Sum of an Infinite Geometric Series



Introduction


An infinite geometric series has the form:


a + a × r + a × r^2 + a × r^3 + ...


= a × (1 + r + r^2 + r^3 + ....)



If |r| < 1, the series converges and a sum exists. 


Σ  a × r^n   =   a ÷ (1 - r)

n=0


Σ  a × r^(n-1)   =   a ÷ (1 - r)

n=1


However if |r| ≥ 1, the series diverges and does not have as sum.



DM42 Program:  GSUM


Calculators: DM42, HP 42S, Free42, Plus42


00 { 36-Byte Prgm }

01 LBL "GSUM"

02 ENTER

03 ABS

04 1

05 X<>Y

06 X>Y?

07 GTO 00

08 R↓

09 R↓

10 1

11 X<>Y

12 -

13 ÷

14 RTN

15 LBL 00

16 CLX

17 "DIVERGES"

18 AVIEW

19 .END.


In the case the series diverges, the X stack is cleared (displays 0).  



TI-84 Plus Program:  GSUM


Calculators:  TI-84 Plus, TI-84 Plus CE (Python), TI-83 Premium CE (Python)


Disp "Σ(A*R^N,0,INF)","Σ(A*R^(N-1),1,I)"

Prompt A,R

ClrHome

Disp "A=",A,"R=",R

If abs(R)<1

Then

A/(1-R)→S

Disp "SUM=",S

Else

Disp "DIVERGES"

End



Examples


A:  3,  R:  -0.2

Stack:  Y = 3, X = -0.2

Result = 2.5


A:  3,  R:  0.2

Stack:  Y = 3, X = -0.2

Result = 3.75


A:  3,  R:  -2.2

Stack:  Y = 3, X = -2.2

Result = diverges


A:  3,  R:  2.2

Stack:  Y = 3, X = 2.2

Result = diverges



Eddie


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