Question

Since the advent of computers, a large number of digits of π has been available. However, before that, many mathematicians found ways to approximate T. For example Machin's formula (1706) is 4 arctan - arctan Leibniz (~1750) found that π can be approximated by: More recently (1914), Srinivasa Ramanujan published dozens of innovative new form ulae for π, remarkable for their elegance, mathematical depth, and rapid convergence. One of his formulae is 1 22 oo (4k)I1103+26390) T 9801kH(3961k) where k!, termed as the factorial of a non-negative integer k, is the product of all positive integers less than or equal to k (note: MATLAB built-in function available) Write a menu-driven program to investigate the constant π. The menu-driven program must have the ollowing options (use menu and sub-menus) Print on screen in a user-friendly way the result from Machin's formul Print on screen in a user-friendly way the approximation of π using Leibniz' formula, allowing the user to specify how many terms to use in the series (n>0). · Print on screen in a user-friendly way the approximation of π using Leibniz' formula, looping until a desired approximation of π is found. The user must specify the acceptable error for the approximation (format: error<10" with 0<n<16 defined by the user). The program print on screen also the number of iterations necessary for achieving the result. · Print on screen in a user-friendly way the approximation of π using Ramanujan's formula allowing the user to specify how many terms to use in the series (0<n<16) Print on screen in a user-friendly way the approximation of π using Ramanujan's formula, looping until a desired approximation is found. The user must specify the acceptable error for the approximation (format: error<10 with 0<n<16 defined by the user). The program prints on screen also the number of iterations necessary for achieving the result. Exit the program. ·

I have completed Machin's formula, but I am having trouble with the other two. This is because the code's for Leibniz's formula and Ramanujan's formula need to be done using loops not syssum or things like that. PLEASE HELP!

Answer 1

ocument Project Iools Browser Emmet Window Help egent-email-proc-monthly billis proc-monthyl billis prod-legent_quotesUntitled1 1 p . @; % Start at 0 2 n input ('How many iterations? Enter a (preferably large) even integer: 5 6 (4*((-1)°1))/(2*ǐ p- p 1); + + % The Leibniz formula for pi: end 8 fprintf('\n%s Iterations: Pi is approximately equal to %1.16f \n",num2str(i),p) 10 12 14 15 16 p-e 17 e input ('enter accecpt able error'); % acceptable error value 18 19 % error <10^-n 20 % n> -log10 (e) 21 n= (-1).* log10(e) 23日if n>-16 24 25 end n-15 26 (4*((-1)"Î))/(2*İ + 1); % The Leibniz formula for pi: 28 29 30 p - p + end jl %1.16f in.,num2str(i).) 31 fprint f('\n%s Iterat ions: Pi is approximately equal to 32 35 36 pi inverse . ; % start at 0 37 n input ( ' (preferably large) even integer; How many iterations? 2.*sqrt (2)/9801 ; Enter a 38 39 41 42 cons - cons . *((factorial(4.*ì)."(1103+26398.*İ))/(factorial(k).zi)."(396."(4..))) p Inverse . end p.1/pilnverse ; fprint f (.\nks 44 45 Pí is approximately equal to %1.16f in.,num2str(),p) 46 47 48 49 iterations:

Document Project Iools Browser Emmet Window Help Legent-email- proc-monthly billis proc-monthyl billi's prod-legent quotesUntitled1 40日for i = 0:n 41 pilnverse . cons . * ( (factoriai(4,4)."(1103+26398 . *İ))/(factori al (k)/i)."(396."(4/i))) 42 end 43 44 p 1/piInverse; 45 46 fprintf 'Inks Iterations: Pi is approximately equal to x1.16f... In', num2str(i),p) 47 48 49 52 53 55 pilnverse = 0; % start at 0 56 e input ('enter accecpt able error'); % acceptable error value 58 % error< 10^-n 59 % n>-log10 (e) 68 n= (-1).* log10(e) 61 62 E3 if n>.16 64 end 66 cons 2.*sqrt (2)/9801; 67日for i = 0:n 68 69 pilnverse- cons . *((factorial(4.*į)."(1103+26398 . *İ))/(factorial (k).^Í)."(396."(4.*i))) end ns.j 71 P 1/piInverseil 7 2 73 74 fprint f('\n%s Iterations: Pį įs approximately equal to %1.16f \n.,num2 str(1),p)