Over the centuries, people have come up with various ways to approximate the num

Question

Over the centuries, people have come up with various ways to approximate the number pi. Here are three infinite series that can be used: Madhava's representation: pi = squareroot 12 sigma^infinity_k=0 (-3)^-k/2k + 1 Newton's: pi = 2 sigma^infinity_k=0 2^k k!^2/(2k + 1)! Ramanujan's: 1/pi = 2 squareroot 2/9801 sigma^infinity_k=0 (4k)! (1103 + 26390k)/(k!)^4 396^4k Do the following: (a) Use the Wikipedia to find when each of the above mathematicians lived. (b) Write script files that sum these series out to the k = 10 term. Find the approximation to pi for each, and compare it to the exact value of pi. How many digits are correct for each? (c) Convert your Newton code to use a while loop instead of a for loop. Use this to determine how many terms must be included in order to approximate pi within 10^-10.

Explanation / Answer

a)

Madhava: 1350 - 1425

Newton : 1643-1727

Ramanujacharya : 1887-1920

b)

Madhava:

approximate value(10 terms) = 3.14159330450308 actual value of pi = 3.14159265358979

5 digits are correct

Newton:

approximate value(10 terms) = 3.14159279510321 actual value of pi = 3.14159265358979

6 digits are correct

Ramanujan:

approximate value(10 terms) = 3.14159265358979 actual value of pi = 3.14159265358979

14 digits are correct

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