Write a program (using a loop) that determines the expression. Run the program with n = 1:10, n=1:1,000, n = 1:100,000. Compare the results with pi function. (use Format long)

The question is incomplete! The complete question along with Matlab code and explanation is provided below!  

Question:

The value of pi can be estimated by: √6*∑1/n² from n=1 to n=∞

Write a program (using a loop) that determines the expression. Run the program with n= 1:10, n=1:1,000, n= 1:100,000. Compare the results with pi function. (use Format long)

Explanation:

We implemented a summation series that approximates the value of pi. The value of pi approaches to the real value of pi as n gets larger. we calculated the value of pi using a for loop with n= 10, n=1,000, and n= 100,000 then compared this value with the value of built-in function of Maltab pi. The results are discussed below.

Matlab Code:

format long

n = 100000;

sum = 0;

for k=1:n

   sum = sum + 1/k^2;

end

value = sqrt(6*sum)

pi

diff = pi - value

Output:

For n = 10:

pi =    3.141592653589793

value =  3.049361635982070

diff =    0.092231017607723

For n = 100:

pi =    3.141592653589793

value =  3.140638056205995

diff =     0.000954597383798

For n = 100,000:

pi =    3.141592653589793

value =    3.141583104326456  

diff =      0.000009549263336

As you can see the difference between real and calculated value of pi is quite large for n = 10 but the difference is very small for n = 100,000. The calculated value is correct up to 4 decimal digits for n = 100,000.