How can I write the approximate value of PI? - c++

Question

the code works but the result keeps getting bigger whenever i put 2 or a higher value in it
#include <iostream>
using namespace std;
int main() {
double pi = 0;
long i;
long n;
cout << "Enter the value of n: ";
cin >> n;
cout << endl;
for (i = 0; i < n; i++)
{
if (i % 2 == 0)
pi = pi + (1 / (2 * i + 1));}
else
pi = pi - (1 / (2 * i + 1));}
pi = 4 * pi;
}
cout << endl << "pi = " << pi << endl;
return 0;
}

Answer 1

Because your { and } is wrong. I think brackets will be as given below
If the formula is PI = 4/1 - 4/3 + 4/5 - 4/7 + ... ( Leibniz's Series ) then you can formalate as given below
#include <iostream>
using namespace std;
int main() {
double n, i; // Number of iterations and control variable
double s = 1; //Signal for the next iteration
double pi = 0.0;
cout << "Enter the value of n: ";
cin >> n;
cout << endl;
cout << "Approximation of the number PI through the Leibniz's series\n";
for(i = 1; i <= (n * 2); i += 2){
pi = pi + s * (4 / i);
s = -s;
cout << "Step (" << (i-1)/2 << "):" << pi << endl;
}
cout << endl << "pi = " << pi << endl;
return 0;
}
Then result will be as if n = 13
Approximation of the number PI through the Leibniz's series
Step (0):4
Step (1):2.66667
Step (2):3.46667
Step (3):2.89524
Step (4):3.33968
Step (5):2.97605
Step (6):3.28374
Step (7):3.01707
Step (8):3.25237
Step (9):3.04184
Step (10):3.23232
Step (11):3.0584
Step (12):3.2184
pi = 3.2184

Answer 2

For Wallis's Series found in Europe by John Wallis in 1655 ( PI = 2/1 x 2/3 x 4/3 x 4/5 x ....) then code will be as
#include <iostream>
using namespace std;
int main()
{
double n, i = 0 ; // Number of iterations and control variable
double pi = 4.;
cout << "Approximation of the number pi through the Wallis's series\n";
cin >> n;
cout << endl;
for(i = 3; i <= (n + 2); i+=2) {
pi = pi * ((i - 1) / i) * (( i + 1) / i);
cout << "Step(" << (i-3)/2 << "):" << pi << endl;
}
cout << "\nAproximated value of PI = " << pi << endl;
}
then result will be
Approximation of the number pi through the Wallis's series
Step(0):3.55556
Step(1):3.41333
Step(2):3.34367
Step(3):3.30239
Step(4):3.2751
Step(5):3.25572
Step(6):3.24125
Step(7):3.23004
Step(8):3.22109
Step(9):3.21378
Step(10):3.20771
Step(11):3.20258
Step(12):3.19818
Step(13):3.19438
Step(14):3.19106
Step(15):3.18813
Step(16):3.18552
Step(17):3.1832
Step(18):3.1811
Step(19):3.17921
Aproximated value of PI = 3.17921

Answer 3

For Nilakantha's Series PI = 3 + 4/(2x3x4) - 4/(4x5x6) + 4/(6x7x8) - ... then code will be as given
#include <iostream>
using namespace std;
int main()
{
double n, i; // Number of iterations and control variable
double s = 1; //Signal for the next operation
double pi = 3;
cout << "Approximation of the number PI through the sequence of the Nilakantha's series\n" ;
cin >> n;
cout << endl;
for(i = 2; i <= n*2; i += 2){
pi = pi + s * (4 / (i * (i + 1) * (i + 2)));
s = -s;
cout << "Step(" << (i-2)/2 << "):" << pi << endl;
}
cout << "\nAproximated value of PI = " << pi << endl;
}
Then result will be as
Approximation of the number PI through the sequence of the Nilakantha's series
Step(0):3.16667
Step(1):3.13333
Step(2):3.14524
Step(3):3.13968
Step(4):3.14271
Step(5):3.14088
Step(6):3.14207
Step(7):3.14125
Step(8):3.14184
Step(9):3.14141
Step(10):3.14174
Step(11):3.14148
Step(12):3.14168
Step(13):3.14152
Step(14):3.14165
Step(15):3.14154
Step(16):3.14164
Step(17):3.14156
Step(18):3.14162
Step(19):3.14157
Aproximated value of PI = 3.14157

Answer 4

Another implementation of the Leibniz's Series:
double pi = 0.0;
double first_denominator = 1.0;
double second_denominator = 3.0;
for (i = 0; i < N; ++i)
{
pi += 4.0 / first_denominator;
pi -= 4.0 / second_denominator;
first_denominator += 2;
second_denominator += 2;
}
By adding in pairs of terms, the need to switch signs has been removed.