Sum of increasing alternate terms in python - evaluation

Question

I need to evaluate the finite sum of numbers which are increasing in absolute value, but are alternate. Problem is: the abolute values grow too fast and it starts accumulating numerical errors. These the functions definitions, one (Gj_full) straight to it and the other (Gj) recursively. fact_quo is a simple factorial function.
def fact_quo(n, m=1):
if (type(n) != int) or (type(m) != int):
raise TypeError("Arguments must be integers.")
if (n < 0) or (m < 0):
raise ValueError("n=" + str(n) + "\t m=" + str(m))
if (n == 0) or (n == 1) or (n == m):
return 1
if (n < m):
raise ValueError("n=" + str(n) + "\t m=" + str(m))
f = n
while n > (m+1):
n -= 1
f *= n
return f
def Gj_full(X, Y, Xl, Yl, j=0, coef=1):
if (X - Y + Xl + Yl) % 2 or X < Y or Y < j:
raise ValueError
u = (X - Y + Xl + Yl) // 2
v = coef * (2 ** (X - Y) * fact_quo(X, Y-j) * fact_quo(u+j, j) *
4 ** j * (-1) ** j)
w = 3 ** (u+j) * fact_quo(X-Y+j)
den2 = fact_quo(X) * fact_quo(Xl) * fact_quo(Yl)
z = (np.sqrt(fact_quo(X)) * np.sqrt(fact_quo(Y))
* np.sqrt(fact_quo(Xl)) * np.sqrt(fact_quo(Yl)))
return (v / (den2 * w)) * z
def Gj(X, Y, Xl, Yl, j=0, coef=1):
if (X - Y + Xl + Yl) % 2 or X < Y or Y < j:
raise ValueError
kX, kY, kXl, kYl, kj = X % 2, Y % 2, Xl % 2, Yl % 2, 0
g = coef * Gj_full(kX, kY, kXl, kYl, kj)
while kX < X:
u = (kX - kY + kXl + kYl) // 2
v = 4 * (u + kj + 1)
w = 3 * (kX + 2 - kY + kj) * (kX + 1 - kY + kj)
g *= (v / w) * np.sqrt(kX + 2) * np.sqrt(kX + 1)
kX += 2
while kXl < Xl:
u = (kX - kY + kXl + kYl) // 2
v = u + kj + 1
w = 3 * (kXl + 2) * (kXl + 1)
g *= (v / w) * np.sqrt(kXl + 2) * np.sqrt(kXl + 1)
kXl += 2
while kYl < Yl:
u = (kX - kY + kXl + kYl) // 2
v = u + kj + 1
w = 3 * (kYl + 2) * (kYl + 1)
g *= (v / w) * np.sqrt(kYl + 2) * np.sqrt(kYl + 1)
kYl += 2
while kY < Y:
u = (kX - kY + kXl + kYl) // 2
v = 3 * (kX - kY + kj) * (kX - kY - 1 + kj)
w = 4 * (kY + 2 - kj) * (kY + 1 - kj) * (u + kj)
g *= (v / w) * np.sqrt(kY + 2) * np.sqrt(kY + 1)
kY += 2
while kj < j:
u = (kX - kY + kXl + kYl) // 2
v = -4 * (kY - kj) * (u + kj + 1)
w = 3 * (kX - kY + kj + 1) * (kj + 1)
g *= (v / w)
kj += 1
return g
The (4/3) ** j and the factorials quicly increase the absolute value of the summing terms. The sum, however, are supposed to be smaller than 1. In fact, for X = Y and Xl = Yl = 0, the sum equals to (-1/3) ** X.

Answer 1

The precision for infinitely large numbers for floats are not available yet without using a lib. Therefore you should look into the decimal lib, you can even set the precision. Eg.
import decimal
decimal.getcontext().prec = 100
def pi():
pi = decimal.Decimal(0)
for k in range(350):
pi += (decimal.Decimal(4)/(decimal.Decimal(8)*decimal.Decimal(k+1))...)
If you manage to force all the numbers to be integers, you don't need to worry about it