| 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)) |