Step by Step solution about How to resolve the algorithm Harmonic series step by step in the C++ programming language

This C++ program generates and prints the harmonic series. The harmonic series is the infinite sum of the reciprocals of the natural numbers:

1 + 1/2 + 1/3 + 1/4 + ...

The program uses the Boost C++ libraries to handle arbitrary-precision integer and rational number arithmetic.

The harmonic_generator class is responsible for generating the terms of the harmonic series. The next() method returns the next term in the series, and the reset() method resets the generator to its initial state.

The main() function first prints the first 20 harmonic numbers. Then, it calculates the 100th harmonic number and prints it. Finally, it finds the positions of the first terms that are greater than 1, 2, ..., 10 and prints those positions.

Here is a breakdown of the key parts of the program:

• The harmonic_generator class:
  • The next() method computes the next term in the harmonic series using the formula:

term_ += 1 / ++n_

where term_ is the current term and n_ is the current natural number.

• The main() function:
  • The first loop prints the first 20 harmonic numbers using the harmonic_generator.
  • The second loop calculates the 100th harmonic number by repeatedly calling the harmonic_generator's next() method.
  • The third loop finds the positions of the first terms that are greater than 1, 2, ..., 10 by comparing the harmonic_generator's next() method's result to the natural number n.

#include <iomanip>
#include <iostream>
#include <boost/rational.hpp>
#include <boost/multiprecision/gmp.hpp>

using integer = boost::multiprecision::mpz_int;
using rational = boost::rational<integer>;

class harmonic_generator {
public:
    rational next() {
        rational result = term_;
        term_ += rational(1, ++n_);
        return result;
    }
    void reset() {
        n_ = 1;
        term_ = 1;
    }
private:
    integer n_ = 1;
    rational term_ = 1;
};

int main() {
    std::cout << "First 20 harmonic numbers:\n";
    harmonic_generator hgen;
    for (int i = 1; i <= 20; ++i)
        std::cout << std::setw(2) << i << ". " << hgen.next() << '\n';
    
    rational h;
    for (int i = 1; i <= 80; ++i)
        h = hgen.next();
    std::cout << "\n100th harmonic number: " << h << "\n\n";

    int n = 1;
    hgen.reset();
    for (int i = 1; n <= 10; ++i) {
        if (hgen.next() > n)
            std::cout << "Position of first term > " << std::setw(2) << n++ << ": " << i << '\n';
    }
}