Lab Test I : Solution

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For students with odd PC numbers

The program

#include <stdio.h>

#define LIMIT 1000000   /* Maximum number of terms */

int main()
{
   double multiplier = 2 * 1.7320508076;  /* 2 * sqrt(3) */
   double threepower;                     /* Power of (-3) for the denominator */
   double sum1, sum2;                     /* sum1 is the partial sum for approximation 1 */
                                          /* sum2 is the partial sum for approximation 2 */
   int n, m;                              /* n is the number of terms. m is for printing. */

   /* Initialize */
   n = 0; m = 10;
   sum1 = sum2 = 1;
   threepower = 1;

   /* Repeat for bigger and bigger values of n */
   while (n <= LIMIT) {
      /* Consider the  next term */
      ++n;

      /* Add (-1)^n / (2n+1) to the first sum */
      if (n%2) sum1 -= 1.0/(double)(2*n+1); else sum1 += 1.0/(double)(2*n+1);

      /* Add 1/[(-3)^n (2n+1)] to the second sum */
      threepower *= (double)(-3);
      sum2 += 1.0/(threepower*(double)(2*n+1));

      /* Check if printing is to be done */
      if (n == m) {
         printf("n = %7d, approximation1 = %12.10lf, approximation2 = %12.10lf\n",
                 n, 4*sum1, multiplier*sum2);

         m *= 10; /* Set value of n for next printing */
      }
   }
}

Output

n =      10, approximation1 = 3.2323158094, approximation2 = 3.1415933046
n =     100, approximation1 = 3.1514934011, approximation2 = 3.1415926536
n =    1000, approximation1 = 3.1425916543, approximation2 = 3.1415926536
n =   10000, approximation1 = 3.1416926436, approximation2 = 3.1415926536
n =  100000, approximation1 = 3.1416026535, approximation2 = 3.1415926536
n = 1000000, approximation1 = 3.1415936536, approximation2 = 3.1415926536

Look how fast the second series has converged, whereas the first series has not converged (with an accuracy of 10 decimal digits) even when one million terms are considered.

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For students with even PC numbers

The program

#include <stdio.h>

#define ERROR 0.00000000001  /* Error bound set at 10^(-11) */

int main ()
{
   double multiplier,        /* The bracketed expression in the n-th term */
          sum, oldsum;       /* The current and the previous sums are maintained
                                for checking convergence within the error limit. */
   int n;                    /* Number of terms */

   /* Initialize */
   sum = multiplier = 0.5;
   oldsum = 0;
   n = 0;

   /* Repeat till the series converges with the desired accuracy */
   while (1) {
      printf("n = %2d, s_n = %12.10lf, approximate value of pi = %12.10lf\n",
              n, sum, 6*sum);

      /* Consider the next term */
      ++n;

      /* Obtain the value of the multiplier for this n from that for the previous n */
      multiplier *= (double)(2*n-1) / (double)(8*n);

      /* Save previous sum */
      oldsum = sum;

      /* Update sum by adding the new term */
      sum += multiplier/((double)(2*n+1));

      /* Check for convergence */
      if (6*(sum-oldsum) < ERROR) exit(0);
   }
}

Output

n =  0, s_n = 0.5000000000, approximate value of pi = 3.0000000000
n =  1, s_n = 0.5208333333, approximate value of pi = 3.1250000000
n =  2, s_n = 0.5231770833, approximate value of pi = 3.1390625000
n =  3, s_n = 0.5235258557, approximate value of pi = 3.1411551339
n =  4, s_n = 0.5235851954, approximate value of pi = 3.1415111723
n =  5, s_n = 0.5235961193, approximate value of pi = 3.1415767158
n =  6, s_n = 0.5235982376, approximate value of pi = 3.1415894253
n =  7, s_n = 0.5235986637, approximate value of pi = 3.1415919824
n =  8, s_n = 0.5235987519, approximate value of pi = 3.1415925112
n =  9, s_n = 0.5235987705, approximate value of pi = 3.1415926229
n = 10, s_n = 0.5235987745, approximate value of pi = 3.1415926469
n = 11, s_n = 0.5235987754, approximate value of pi = 3.1415926521
n = 12, s_n = 0.5235987755, approximate value of pi = 3.1415926533
n = 13, s_n = 0.5235987756, approximate value of pi = 3.1415926535
n = 14, s_n = 0.5235987756, approximate value of pi = 3.1415926536
n = 15, s_n = 0.5235987756, approximate value of pi = 3.1415926536

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