Casio fx-9860g and fx-CG 50: Binomial Series
Generating The Binomial Series
The binomial series (1 + b*x)^a can be generated by the series:
(1 + b*x)^a = 1 + a*b*x + a*(a-1) / 2! * (b*x)^2 + a*(a-1)*(a-2) / 3! * (b*x)^3 + ....
= ∑ ( a NCR k) * (b * x)^k for k = 0 from ∞
where:
(a NCR k) = (a * (a - 1) * (a - 2) * ... * (a - k + 1) ) / k!
The values a and b can be complex and do not have to be integers.
If a is not a positive integer, the series continues on indefinitely.
Output: List 5 has the coefficients. The program BINOMSRS calculates the coefficients and any approximation of that series (f(x)).
Casio fx-9860GII and fx-CG 50 Program BINOMSRS
Lbl 2
"2020-02-24 EWS"
"BINOMIAL SERIES"
"EXPAND (1+BX)^A"
"B"? → B
"A"? → A
Lbl 0
"TERMS (≥3)"? → T
Int T → T
T < 3 ⇒ Goto 0
T → Dim List 5
1 → List 5[1]
1 → N
For 2 → K To T
N (A - (K - 2)) → N
N ÷ (K - 1)! * B^(K - 1) → List 5[K]
Next
Lbl 1
Menu "MENU","LIST COEFS",C,"APPROX F(X)",E,"NEW SERIES",2,"EXIT",X
Lbl C
"COEFS IN List 5"
List 5 ◢
Goto 1
Lbl E
"X"? → X
Sum( List 5 * X^Seq(I, I, 0, T-1, 1) ) → Y
"F(X) IS ABOUT:"
Y ◢
Goto 1
Lbl X
"DONE"
The program allows the user to evaluate the series for given values and create new series for different problems.
Download the fx-9860gII and fx-9750gII version:
https://drive.google.com/open?id=1fgxjwf4UkyxhY7XIWqx7HRQHJf-PKnbi
Download the fx-CG 50 and fx-CG 10/20 version:
https://drive.google.com/open?id=1H0Sq2-h6NcWFIw7_pM25PAHHQ2QKtKEy
Example
Expand (1 - x)^(1/2) to six terms. Approximate f(0.35).
Hint: Use the fraction key ( [ a b/c ] or [ []/[] ] ) to get results in fractions (whenever possible).
b = -1
a = 1/2
t = 6
Coefficients (List 5):
{1, -1/2, -1/8, -1/16, -5/128, -7/256}
(1 - x)^(1/2) ≈ 1 - 1/2 * x - 1/8 * x^2 - 1/16 * x^3 - 5/128 * x^4 - 7/256 * x^5
Approx F(X): X = 0.35
Result: 0.8062780164
Eddie
All original content copyright, © 2011-2020. Edward Shore. Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited. This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.
Generating The Binomial Series
The binomial series (1 + b*x)^a can be generated by the series:
(1 + b*x)^a = 1 + a*b*x + a*(a-1) / 2! * (b*x)^2 + a*(a-1)*(a-2) / 3! * (b*x)^3 + ....
= ∑ ( a NCR k) * (b * x)^k for k = 0 from ∞
where:
(a NCR k) = (a * (a - 1) * (a - 2) * ... * (a - k + 1) ) / k!
The values a and b can be complex and do not have to be integers.
If a is not a positive integer, the series continues on indefinitely.
Output: List 5 has the coefficients. The program BINOMSRS calculates the coefficients and any approximation of that series (f(x)).
Casio fx-9860GII and fx-CG 50 Program BINOMSRS
Lbl 2
"2020-02-24 EWS"
"BINOMIAL SERIES"
"EXPAND (1+BX)^A"
"B"? → B
"A"? → A
Lbl 0
"TERMS (≥3)"? → T
Int T → T
T < 3 ⇒ Goto 0
T → Dim List 5
1 → List 5[1]
1 → N
For 2 → K To T
N (A - (K - 2)) → N
N ÷ (K - 1)! * B^(K - 1) → List 5[K]
Next
Lbl 1
Menu "MENU","LIST COEFS",C,"APPROX F(X)",E,"NEW SERIES",2,"EXIT",X
Lbl C
"COEFS IN List 5"
List 5 ◢
Goto 1
Lbl E
"X"? → X
Sum( List 5 * X^Seq(I, I, 0, T-1, 1) ) → Y
"F(X) IS ABOUT:"
Y ◢
Goto 1
Lbl X
"DONE"
The program allows the user to evaluate the series for given values and create new series for different problems.
Download the fx-9860gII and fx-9750gII version:
https://drive.google.com/open?id=1fgxjwf4UkyxhY7XIWqx7HRQHJf-PKnbi
Download the fx-CG 50 and fx-CG 10/20 version:
https://drive.google.com/open?id=1H0Sq2-h6NcWFIw7_pM25PAHHQ2QKtKEy
Example
Expand (1 - x)^(1/2) to six terms. Approximate f(0.35).
Hint: Use the fraction key ( [ a b/c ] or [ []/[] ] ) to get results in fractions (whenever possible).
b = -1
a = 1/2
t = 6
Coefficients (List 5):
{1, -1/2, -1/8, -1/16, -5/128, -7/256}
(1 - x)^(1/2) ≈ 1 - 1/2 * x - 1/8 * x^2 - 1/16 * x^3 - 5/128 * x^4 - 7/256 * x^5
Approx F(X): X = 0.35
Result: 0.8062780164
Eddie
All original content copyright, © 2011-2020. Edward Shore. Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited. This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.