The Euler Series

# "The Euler Series" p.31 in "The Pleasures of Pi,e"

# 1/1**2 + 1/2**2 + 2/3**2 + ... + 1/n**2 + ... = pi**/6 (limit)  (very slow to converge)

# So I use Hans Lundmark's post on http://math.stackexchange.com, at
# http://math.stackexchange.com/questions/8337/different-methods-to-compute-sum-n-1-infty-frac1n2

from mpmath import *; mp.dps = 100; mp.pretty = True

def compareSequences(seq1, seq2):
    """
    Returns first index at which two sequences differ

    >>> seq1 = "Now is the time"
    >>> seq2 = "Now was the time"
    >>> compareSequences(seq1, seq2)
    4
    """
    for index in range(len(seq2)):
        if seq1[index] != seq2[index]:
            return index
    return None

def int_commas(n):
    """
    inserts commas into integers and returns the string

    E.g. -12345678 -> -12,345,789
    """
    return format(n, ',d')



n = mpf(150)
print("n =", int_commas(int(n)))

print()
limit = (pi()**mpf(2))/mpf(6)
upper_bound_partial_sum = (mpf(2)*(4**n - 1)/mpf(3)) * (pi()**mpf(2)/(4**(n+1)))
idx1 = compareSequences(str(upper_bound_partial_sum), str(limit))

lower_bound_partial_sum = (mpf(2)*(4**n - 1)/mpf(3) - (mpf(2)**n - 1)) * (pi()**mpf(2)/(4**(n+1)))
idx2 = compareSequences(str(lower_bound_partial_sum), str(limit))

print(limit, "limit (PI**2/6)")
print((idx1-1)*' ', '^')
print(upper_bound_partial_sum, "upper_bound_partial_sum")
print()
print()
print(limit, "limit (PI**2/6)")
print((idx2-1)*' ', '^')
print(lower_bound_partial_sum, "lower_bound_partial_sum")

"""
OUTPUT
n = 150

1.644934066848226436472415166646025189218949901206798437735558229370007470403200873833628900619758705 limit (PI**2/6)
                                                                                         ^
1.644934066848226436472415166646025189218949901206798437735558229370007470403200873833628899812245197 upper_bound_partial_sum


1.644934066848226436472415166646025189218949901206798437735558229370007470403200873833628900619758705 limit (PI**2/6)
                                            ^
1.644934066848226436472415166646025189218949899478015751393061460363041277434208815632129055954370542 lower_bound_partial_sum
"""