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848d4b7 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 | from mpmath import mp
mp.dps = 110
def compute():
# S_5 = sum_{n>=1} 1/(n^5 * binom(2n,n))
# Use recurrence for a_n = 1/binom(2n,n): a_{n+1} = a_n * (n+1)/(4n+2)
target = mp.eps * mp.mpf('1e-20')
r_upper = mp.mpf('0.251') # safely above the true term ratio (< 1/4)
s = mp.mpf('0')
a = mp.mpf('0.5') # a_1 = 1/binom(2,1)
n = 1
while True:
t = a / (n**5)
s += t
# remainder bound assuming geometric ratio <= r_upper:
# R_n = sum_{k>=1} t_{n+k} <= t_n * r_upper/(1-r_upper)
if t * r_upper / (1 - r_upper) < target:
break
a *= mp.mpf(n + 1) / mp.mpf(4 * n + 2)
n += 1
if n > 200000:
raise RuntimeError("Convergence failure")
return s
if __name__ == "__main__":
print(mp.nstr(compute(), mp.dps)) |