Homework Help is Here – Start Your Trial Now!
Engineering
Computer Engineering
(a) The Gregory-Leibniz formula for 7 dates from the 1670s (but was apparently known to Indian mathematician Madhava of Sangamagrama around 1400), and states that ㅠ 4 = 1 1 1 1 + + 3 5 7 9 1 + -1)². 1 2i+1 +... Write a function pi_gl(n) which takes as an input an integer n and returns an approximation of , by evaluating the Gregory-Leibniz formula with n terms. (For n=5 your function should return 3.3396825396.... Don't forget to multiply the sum of the series by 4.)

(a) The Gregory-Leibniz formula for 7 dates from the 1670s (but was apparently known to Indian mathematician Madhava of Sangamagrama around 1400), and states that ㅠ 4 = 1 1 1 1 + + 3 5 7 9 1 + -1)². 1 2i+1 +... Write a function pi_gl(n) which takes as an input an integer n and returns an approximation of , by evaluating the Gregory-Leibniz formula with n terms. (For n=5 your function should return 3.3396825396.... Don't forget to multiply the sum of the series by 4.)

Computer Networking: A Top-Down Approach (7th Edition)
Computer Networking: A Top-Down Approach (7th Edition)
7th Edition
ISBN:9780133594140
Author:James Kurose, Keith Ross
Publisher:James Kurose, Keith Ross
Chapter1: Computer Networks And The Internet
Section: Chapter Questions
Problem R1RQ: What is the difference between a host and an end system? List several different types of end...
See similar textbooks
icon
Related questions
Question
(a) The Gregory-Leibniz formula for 7 dates from the 1670s (but was apparently known to Indian mathematician Madhava of Sangamagrama around 1400), and states that ㅠ 4 ㅠ 2 π=3+ = 3 + - 1 2 2 X 1 - 1 Write a function pi_gl(n) which takes as an input an integer n and returns an approximation of , by evaluating the Gregory-Leibniz formula with n terms. (For n=5 your function should return 3.3396825396.... Don't forget to multiply the sum of the series by 4.) (b) Another series expression for π, less well-known and discovered by Nikalantha (around 1500), is X + 3 5 4 2 × 3 × 4 1 3 4(−1)²+1 2i (2i + 1)(2i + 2) X 1 1 + 7 9 4 4× 5 × 6 4 5 Write a function pi_nik(n) which takes as an input an integer n and returns an approximation of , by evaluating this formula with n terms. (For n=3 your function should return 3.1333333....) (c) A third expression is due to Wallis (1656): X + 6 5 + -1) ². + ... 1 2i+1 X 4 4 6 × 7 ×8 8 × 9 × 10 +... X... X 2i 2i X 2i 1 2i + 1 X... Write a function pi_wallis (n) which takes an integer n as input and returns an approximation of by evaluating the Wallis formlua with n brackets in the formula. (For example, pi_wallis (3) should return 2.9257142857....) (d) For the Gregory-Leibniz expression, compute the absolute value of the error (that is, abs (pi_gl(n)- math.pi)) for n = 1 to 500. Store these errors in a list (say, pi_gl_error), and repeat this for the other two series, storing the errors in new lists. Plot a graph with the errors pi_gl_error, pi_nik_error pi_wallis_error on the y-axis, against the range (1,500) on the x-axis. (You should plot all three lines on the same graph. Be sure to label axes and data. You should choose for yourself whether plot, loglog or semilogy gives the most meaningful plot.) (e) Give a brief comment on the rate at which each expression converges to T
Transcribed Image Text:(a) The Gregory-Leibniz formula for 7 dates from the 1670s (but was apparently known to Indian mathematician Madhava of Sangamagrama around 1400), and states that ㅠ 4 ㅠ 2 π=3+ = 3 + - 1 2 2 X 1 - 1 Write a function pi_gl(n) which takes as an input an integer n and returns an approximation of , by evaluating the Gregory-Leibniz formula with n terms. (For n=5 your function should return 3.3396825396.... Don't forget to multiply the sum of the series by 4.) (b) Another series expression for π, less well-known and discovered by Nikalantha (around 1500), is X + 3 5 4 2 × 3 × 4 1 3 4(−1)²+1 2i (2i + 1)(2i + 2) X 1 1 + 7 9 4 4× 5 × 6 4 5 Write a function pi_nik(n) which takes as an input an integer n and returns an approximation of , by evaluating this formula with n terms. (For n=3 your function should return 3.1333333....) (c) A third expression is due to Wallis (1656): X + 6 5 + -1) ². + ... 1 2i+1 X 4 4 6 × 7 ×8 8 × 9 × 10 +... X... X 2i 2i X 2i 1 2i + 1 X... Write a function pi_wallis (n) which takes an integer n as input and returns an approximation of by evaluating the Wallis formlua with n brackets in the formula. (For example, pi_wallis (3) should return 2.9257142857....) (d) For the Gregory-Leibniz expression, compute the absolute value of the error (that is, abs (pi_gl(n)- math.pi)) for n = 1 to 500. Store these errors in a list (say, pi_gl_error), and repeat this for the other two series, storing the errors in new lists. Plot a graph with the errors pi_gl_error, pi_nik_error pi_wallis_error on the y-axis, against the range (1,500) on the x-axis. (You should plot all three lines on the same graph. Be sure to label axes and data. You should choose for yourself whether plot, loglog or semilogy gives the most meaningful plot.) (e) Give a brief comment on the rate at which each expression converges to T
Expert Solution
steps

Step by step

Solved in 4 steps with 2 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
Computer Networking: A Top-Down Approach (7th Edi…
Computer Networking: A Top-Down Approach (7th Edi…
Computer Engineering
ISBN:
9780133594140
Author:
James Kurose, Keith Ross
Publisher:
PEARSON
Computer Organization and Design MIPS Edition, Fi…
Computer Organization and Design MIPS Edition, Fi…
Computer Engineering
ISBN:
9780124077263
Author:
David A. Patterson, John L. Hennessy
Publisher:
Elsevier Science
Network+ Guide to Networks (MindTap Course List)
Network+ Guide to Networks (MindTap Course List)
Computer Engineering
ISBN:
9781337569330
Author:
Jill West, Tamara Dean, Jean Andrews
Publisher:
Cengage Learning
Concepts of Database Management
Concepts of Database Management
Computer Engineering
ISBN:
9781337093422
Author:
Joy L. Starks, Philip J. Pratt, Mary Z. Last
Publisher:
Cengage Learning
Prelude to Programming
Prelude to Programming
Computer Engineering
ISBN:
9780133750423
Author:
VENIT, Stewart
Publisher:
Pearson Education
Sc Business Data Communications and Networking, T…
Sc Business Data Communications and Networking, T…
Computer Engineering
ISBN:
9781119368830
Author:
FITZGERALD
Publisher:
WILEY
SEE MORE TEXTBOOKS