class Fraction:
    #Constructor. Puts fraction in simplest form
    def __init__(self,a,b):
        self.num = a
        self.den = b
        self.simplify()
    #Print Fraction as a String
    def __str__(self):
        if self.den==1:
            return str(self.num)
        else:
            return str(self.num)+"/"+str(self.den)
    #Get the Numerator
    def getNum(self):
        return self.num
    #Get the Denominator
    def getDen(self):
        return self.den
    #Give Numerical Approximation of Fraction
    def approximate(self):
        return self.num/self.den
    #Simplify fraction
    def simplify(self):
        x = self.gcd(self.num,self.den)
        self.num = self.num // x
        self.den = self.den // x
    #Find the GCD of a and b
    def gcd(self,a,b):
        if b==0:
            return a
        else:
            return self.gcd(b,a % b)
    #Complete these methods in lab
    def __add__(self,other):
      newnum = self.num*other.den + self.den*other.num
      newden = self.den*other.den
      common = self.gcd(newnum, newden)
      return Fraction(newnum//common, newden//common)
    def __sub__(self,other):
      newnum = self.num*other.den - self.den*other.num
      newden = self.den*other.den
      common = self.gcd(newnum, newden)
      return Fraction(newnum//common, newden//common)
    def __mul__(self,other):
        newnum = self.num*other.num
        newden = self.den*other.den
        return Fraction(newnum, newden)
    def __truediv__(self,other):
      newnum = self.num * other.den
      newden = self.den * other.num
      return Fraction(newnum, newden)
    def __pow__(self,exp):
            if exp == 0:
                return 1
            elif exp > 0:
                return Fraction(self.num**exp, self.den**exp)
            else:
                return Fraction(self.den**abs(exp), self.num**abs(exp))

# '''x = Fraction(1,2)
# y = Fraction(2,3)
# print(x+y)
# print(x-y)
# print(x*y)
# print(x/y)
# print(x**2)
# print(y**2)'''

print('Welcome to Fun with Fractions!')
number = input('Enter Number of iterations (integer>0):\n')
while True:
        number = int(number)
        if number >= 0:
            break
        else:
            number = input('Enter Number of iterations (integer>0):\n')
harmony = Fraction(0,1)
for k in range(1, (number+1)):
    harmony += Fraction(1, k)
print('H' + '(' + str(number) + ')=' + str(harmony))
print('H' + '(' + str(number) + ')~=' + str(harmony.approximate()))

two = Fraction(0,1)
for k in range(0, (number+1)):
    two += Fraction(1,2**k)
print('T' + '(' + str(number) + ')=' + str(two))
print('T' + '(' + str(number) + ')~=' + str(two.approximate()))

zero = Fraction(0,1)
for k in range(0, (number+1)):
    two += Fraction(1,2**k)
zero_result = Fraction(2,1) - two
print('Z' + '(' + str(number) + ')=' + str(zero_result))
print('Z' + '(' + str(number) + ')~=' + str(zero_result.approximate()))

def Rieman(number, b):
    riemann = Fraction(0,1)
    for k in range(1, (number+1)):
        riemann += (Fraction(1, k)**b)
    return riemann
for k in range(2,9):
    print('R' + '(' + str(number) + ',' + str(k) + ')=' + str(Rieman(number, k)))
    print('R' + '(' + str(number) + ',' + str(k) + ')~=' + str(Rieman(number, k).approximate()))