Рекурся

double sum = 1 ;
double a = x;
for ( int i = 1 ; i < n; i++)
{
     a *= x/i;
     sum +=a;
}

#include <iostream>
 
double e(double);
double e_helper(double, double, double, size_t);
 
int main()
{
    double x;
 
    std::cout << "Enter x: ";
    std::cin >> x;
 
    std::cout << "Result = " << e(x) << std::endl;
}
 
double e(double x)
{
    return e_helper(x, 1E-8, 1.0, 1);
}
 
double e_helper(double x, double eps, double an, size_t n)
{
    return (an < eps) ? 0.0 : an + e_helper(x, eps, an * x / n, n + 1);
}

#include <iostream>
 
double e(double, size_t);
double e_helper(double, size_t, double, size_t);
 
int main()
{
    double x;
    size_t N;
 
    std::cout << "Enter x: ";
    std::cin >> x;
    std::cout << "Enter N: ";
    std::cin >> N;
 
    std::cout << "Result = " << e(x, N) << std::endl;
}
 
double e(double x, size_t N)
{
    return e_helper(x, N, 1.0, 1);
}
 
double e_helper(double x, size_t N, double an, size_t n)
{
    return (n == N) ? 0.0 : an + e_helper(x, N, an * x / n, n + 1);
}