How to calculate the value of pi accuratly using while loops

[rrs274]

I need to calculate the value of pi accurately to six digits using the series pi=4-4/3+4/5-4/7+4/9-4/11... Honestly I have no idea how they want me to do with this. I know how to express a while loop but I haven't had any experience with the series they have given. Please help if you can, thanks:)

[kev82]

Maybe this will help

The i'th term of that sequence is given by the formula:

(-1)^(i+1) * 4/(2*i-1)

You want to repeatedly keep adding the next term until the total is correct to 6dp. I think that would be

fabs(total - 3.141592d) < 0.0000005d

[Alrededor]

The series comes from the power series of Arctan(x) = x - x³/3 + x⁵/5 - x⁷/7 ... with x = 1 yielding
Pi / 4 = 1 - 1/3 + 1/5 - 1/7 ...
Multiplying by 4 gives the series you show above. Notice that the numerator in each new term is always 4 and the denominator is always 2 more than the previous term so using a while loop something along the line of this should work.

double total = 0;
long int dem = 1;
bool isPositive = true;
while ( fabs(total - 3.141592d) < 0.0000005d )
{
   if ( isPositive )
   {
      total += 4.0 / dem;
   }
   else
   {
      total -= 4.0 / dem;
   }
   isPositive = !isPositive;
  dem += 2;
}

Notice that using a while loop requires you to already know pi accurately to test against your running total in the condition of the loop. A more sophisticated approach could use a for loop by analyzing the series itself to figure out how many terms to add. Consider the following theorem from calculus:

If for the series ∑a_n a_n→ 0 as n→infinity and the terms alternate in sign and decrease in magnitude monotonically to zero, i.e., |a_n| > |a_n+1| then the series converges and the error in the partial sum obtained by truncating the series is less in magnitude than the first term omitted. That is,

if S = ∑a_n and P_n = a₀ + a₁ + a₂ + ... + a_n then the magnitude of the error = | S - P_n | < | a_n+1 |

As noted by kev82 the general term has the form (-1)⁽ⁱ⁺¹⁾ * 4/(2*i-1) with the index i starting at i=1 so what we want is

| S - P_n | < | a_n+1 | = | (-1)⁽ⁿ⁺²⁾ * 4/(2*n+1) | < .000001
where I am starting the index at zero.

Simplifying we have

4 / (2*n + 1) < .000001
4 / .000001 < 2 * n + 1
4000000 < 2 * n + 1
3999999 < 2*n
1999999.5 < n

Thus if we were to add two million and one terms of the series we should have the desired accuracy.