Summation & Product iterated operators

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Below is the code for JAVA, the code for C# is almost identical.

Case 1: SIGMA summation operator

$$\sum_{i=k}^{n}f(x_1,x_2,,\ldots,i)$$

import org.mariuszgromada.math.mxparser.*;
...
Expression e1 = new Expression("sum(i, 1, 10, 2*i)");
mXparser.consolePrintln("Res 1: " + e1.getExpressionString() + " = " + e1.calculate());
         
/* Iteration can be done by not necessarily whole increment */
Expression e2= new Expression("sum(i, 1, 10, i, 0.5)");
mXparser.consolePrintln("Res 2: " + e2.getExpressionString() + " = " + e2.calculate());

[mXparser-v.5.0.0] Res 1: sum(i, 1, 10, 2*i) = 110.0
[mXparser-v.5.0.0] Res 2: sum(i, 1, 10, i, 0.5) = 104.5

Case 2: PI product operator

$$\prod_{i=k}^{n}f(x_1,x_2,,\ldots,i)$$

import org.mariuszgromada.math.mxparser.*;
...
/* factorial */
Expression e1 = new Expression("prod(i, 1, 5, i)");
mXparser.consolePrintln("Res 1: " +  e1.getExpressionString() + " = " + e1.calculate());
         
/* Iteration can be done by not necessarily whole increment */
/* Here different form of 10! */
Expression e2 = new Expression("prod(i, 1, 5, 2*i, 0.5)");
mXparser.consolePrintln("Res 2: " + e2.getExpressionString() + " = " + e2.calculate());

[mXparser-v.5.0.0] Res 1: prod(i, 1, 5, i) = 120.0
[mXparser-v.5.0.0] Res 2: prod(i, 1, 5, 2*i, 0.5) = 3628800.0

Case 3: SIGMA summation operator – Approximating sin(x) by Taylor series

import org.mariuszgromada.math.mxparser.*;
...
Argument x = new Argument("x = 2*pi");
Argument n = new Argument("n = 3");
Expression e = new Expression("sin(x) - sum(k, 0, n, (-1)^k*(x^(2*k+1))/(2*k+1)! )", x, n);
mXparser.consolePrintln("Res 1: " + e.getExpressionString() + " = " + e.calculate());
 
/* Increasing polynomial order ... */
n.setArgumentValue(5);
mXparser.consolePrintln("Res 2: " + e.getExpressionString() + " = " + e.calculate());
 
/* Increasing polynomial order ... */
n.setArgumentValue(10);     
mXparser.consolePrintln("Res 3: " + e.getExpressionString() + " = " + e.calculate());
         
/* Increasing polynomial order ... */
n.setArgumentValue(20);     
mXparser.consolePrintln("Res 4: " + e.getExpressionString() + " = " + e.calculate());
         
/* Checking other point closer to '0' */
x.setArgumentValue(2);
mXparser.consolePrintln("Res 5: " + e.getExpressionString() + " = " + e.calculate());

[mXparser-v.5.0.0] Res 1: sin(x) - sum(k, 0, n, (-1)^k*(x^(2*k+1))/(2*k+1)! ) = 30.159127410206484
[mXparser-v.5.0.0] Res 2: sin(x) - sum(k, 0, n, (-1)^k*(x^(2*k+1))/(2*k+1)! ) = 3.195076042131831
[mXparser-v.5.0.0] Res 3: sin(x) - sum(k, 0, n, (-1)^k*(x^(2*k+1))/(2*k+1)! ) = -8.27409522186923E-5
[mXparser-v.5.0.0] Res 4: sin(x) - sum(k, 0, n, (-1)^k*(x^(2*k+1))/(2*k+1)! ) = 0.0
[mXparser-v.5.0.0] Res 5: sin(x) - sum(k, 0, n, (-1)^k*(x^(2*k+1))/(2*k+1)! ) = 0.0

Case 4: SIGMA summation operator – Approximating pi value by integrating sqrt(1-x^2)

import org.mariuszgromada.math.mxparser.*;
...
Argument d = new Argument("d",0.1);
Expression e = new Expression("2 * sum(x, -1, 1, d*sqrt(1-x^2), d)", d);
mXparser.consolePrintln("Res 1: d = " + d.getArgumentValue() + ", " + e.getExpressionString() + " = " + e.calculate());
d.setArgumentValue(0.01);
mXparser.consolePrintln("Res 2: d = " + d.getArgumentValue() + ", " + e.getExpressionString() + " = " + e.calculate());

[mXparser-v.5.0.0] Res 1: d = 0.1, 2 * sum(x, -1, 1, d*sqrt(1-x^2), d) = 3.1045183304630037
[mXparser-v.5.0.0] Res 2: d = 0.01, 2 * sum(x, -1, 1, d*sqrt(1-x^2), d) = 3.1404170317790436