Step by Step solution about How to resolve the algorithm Almkvist-Giullera formula for pi step by step in the Mathematica/Wolfram Language programming language

Purpose:

This Wolfram programming language code calculates a mathematical expression involving factorials and binomial coefficients, computes its sum over a range of values, and then calculates the reciprocal square root of the sum.

Detailed Explanation:

• Step 1: Define Functions:

  • numerator[n_] defines the numerator of a mathematical expression involving factorials and binomial coefficients.
  • denominator[n_] defines the denominator of the expression.

• Step 2: Evaluate Numerator:

  • The expression numerator /@ Range[0, 9] evaluates the numerator for values of n from 0 to 9 and stores the results in a list.

• Step 3: Calculate Sum and Compute Value:

  • The expression val = 1/Sqrt[Total[numerator[#]/denominator[#] & /@ Range[0, 100]]]; calculates the sum of the numerator/denominator ratio from 0 to 100 and assigns the square root reciprocal of the sum to the variable val.

• Step 4: Display the Result:

  • The expression N[val, 70] displays the numeric approximation of val with 70 decimal places.

Output:

The code will output a numeric approximation of the value of val, which is a complex number with both real and imaginary components. The specific value will depend on the accuracy of the calculation and the precision specified in the N function.

ClearAll[numerator, denominator]
numerator[n_] := (2^5) ((6 n)!) (532 n^2 + 126 n + 9)/(3 (n!)^6)
denominator[n_] := 10^(6 n + 3)
numerator /@ Range[0, 9]
val = 1/Sqrt[Total[numerator[#]/denominator[#] & /@ Range[0, 100]]];
N[val, 70]