Question

The mathematical constant Pi is an irrational number with value approximately 3.1415928... The precise value of this constant can be obtained from the following infinite sum:
Pi^2 = 8+8/3^2+8/5^2+8/7^2+8/9^2+...
(Pi is of course just the square root of this value.)
Although we cannot compute the entire infinite series, we get a good approximation of the value of Pi' by computing the beginning of such a sum. Write a function approxPIsquared that takes as input float error and approximates constant Pi to within error by computing the above sum, term by term, until the difference between the new and the previous sum is less than error. The function should return the new sum
>>>approxPIsquared(0.0001)
9.855519952254232
>>>approxPIsquared(0.00000001)
9.869462988376474

Answer 1

Answer:

I am writing a Python program:

def approxPIsquared(error):

   previous = 8

   new_sum =0

   num = 3

   while (True):

       new_sum = (previous + (8 / (num ** 2)))

       if (new_sum - previous <= error):

           return new_sum

       previous = new_sum

       num+=2    

print(approxPIsquared(0.0001))

Explanation:

I will explain the above function line by line.

def approxPIsquared(error):  

This is the function definition of approxPlsSquared() method that takes error as its parameter and approximates constant Pi to within error.

previous = 8     new_sum =0      num = 3

These are variables. According to this formula:

Pi^2 = 8+8/3^2+8/5^2+8/7^2+8/9^2+...

Value of previous is set to 8 as the first value in the above formula is 8. previous holds the value of the previous sum when the sum is taken term by term. Value of new_sum is initialized to 0 because this variable holds the new value of the sum term by term. num is set to 3 to set the number in the denominator. If you see the 2nd term in above formula 8/3^2, here num = 3. At every iteration this value is incremented by 2 to add 2 to the denominator number just as the above formula has 5, 7 and 9 in denominator.

while (True):  This while loop keeps repeating itself and calculates the sum of the series term by term, until the difference between the value of new_sum and the previous is less than error. (error value is specified as input).

new_sum = (previous + (8 / (num ** 2)))  This statement represents the above given formula. The result of the sum is stored in new_sum at every iteration. Here ** represents num to the power 2 or you can say square of value of num.

if (new_sum - previous <= error):  This if condition checks if the difference between the new and previous sum is less than error. If this condition evaluates to true then the value of new_sum is returned. Otherwise continue computing the, sum term by term.

return new_sum  returns the value of new_sum when above IF condition evaluates to true

previous = new_sum  This statement sets the computed value of new_sum to the previous.

For example if the value of error is 0.0001 and  previous= 8 and new_sum contains the sum of a new term i.e. the sum of 8+8/3^2 = 8.88888... Then IF condition checks if the

new_sum-previous <= error

8.888888 - 8 = 0.8888888

This statement does not evaluate to true because 0.8888888... is not less than or equal to 0.0001

So return new_sum statement will not execute.

previous = new_sum statement executes and now value of precious becomes 8.888888...

Next   num+=2  statement executes which adds 2 to the value of num. The value of num was 3 and now it becomes 3+2 = 5.

After this while loop execute again computing the sum of next term using       new_sum = (previous + (8 / (num ** 2)))  

new_sum = 8.888888.. + (8/(5**2)))

This process goes on until the difference between the new_sum and the previous is less than error.

screenshot of the program and its output is attached.