Pi greco con serie di Eulero – Programmare con Python

La serie

$\displaystyle \frac{\pi^2}{6}=\sum_{n=1}^{+\infty}\frac{1}{n^2}=\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\dots=1+\frac{1}{4}+\frac{1}{9}+\dots$

import math

def a(n): return 1/n**2

PASSI = 20
S_n   = 0
for n in range(1,PASSI+1):
    a_n   = a(n)
    S_n  += a_n
    pi_n  = math.sqrt(6*S_n)

    print("%02i | %12.10f | %12.10f | %12.10f" %(n, a_n, S_n, pi_n))

Output

01 | 1.0000000000 | 1.0000000000 | 2.4494897428
02 | 0.2500000000 | 1.2500000000 | 2.7386127875
03 | 0.1111111111 | 1.3611111111 | 2.8577380332
04 | 0.0625000000 | 1.4236111111 | 2.9226129861
05 | 0.0400000000 | 1.4636111111 | 2.9633877010
06 | 0.0277777778 | 1.4913888889 | 2.9913764947
07 | 0.0204081633 | 1.5117970522 | 3.0117739478
08 | 0.0156250000 | 1.5274220522 | 3.0272978567
09 | 0.0123456790 | 1.5397677312 | 3.0395075896
10 | 0.0100000000 | 1.5497677312 | 3.0493616360
11 | 0.0082644628 | 1.5580321940 | 3.0574815067
12 | 0.0069444444 | 1.5649766384 | 3.0642878178
13 | 0.0059171598 | 1.5708937982 | 3.0700753719
14 | 0.0051020408 | 1.5759958390 | 3.0750569156
15 | 0.0044444444 | 1.5804402834 | 3.0793898260
16 | 0.0039062500 | 1.5843465334 | 3.0831930203
17 | 0.0034602076 | 1.5878067411 | 3.0865580258
18 | 0.0030864198 | 1.5908931608 | 3.0895564350
19 | 0.0027700831 | 1.5936632439 | 3.0922450523
20 | 0.0025000000 | 1.5961632439 | 3.0946695241

matplotlib

import math                     # sqrt
import matplotlib.pyplot as plt # ...
 
def a(n): return 1/n**2

PASSI = 10
N     = []
A     = []
PI    = []
S_n   = 0
for n in range(1, PASSI+1):
    a_n   = a(n)
    S_n  += a_n
    pi_n  = math.sqrt(6*S_n)

    print(n, a_n, S_n, pi_n)

    N.append(n)
    A.append(a_n)
    PI.append(pi_n)

plt.subplot(2, 1, 1) 
plt.grid()
plt.plot(N, PI, linewidth="2")
plt.title("Pi greco: serie di Eulero")
plt.ylim(0,3.5)
plt.ylabel("pi(n)")

plt.subplot(2, 1, 2) 
plt.bar(N, A, color="green")
plt.grid()
plt.xlabel("n")
plt.ylabel("a(n)")

plt.show()

fractions

Utilizzando il modulo fractions si ottengono le approssimazioni successive come frazioni!

import fractions
import math

def a(n): return fractions.Fraction(1, n**2)

PASSI = 20
S_n   = 0
for n in range(1, PASSI+1):
    a_n  = a(n)
    S_n += a_n
    pi_n = math.sqrt(6*S_n)
    print("%02i | %5s | %35s | %12.10f" %(n, a_n, S_n, pi_n))

Output

01 |     1 |                                   1 | 2.4494897428
02 |   1/4 |                                 5/4 | 2.7386127875
03 |   1/9 |                               49/36 | 2.8577380332
04 |  1/16 |                             205/144 | 2.9226129861
05 |  1/25 |                           5269/3600 | 2.9633877010
06 |  1/36 |                           5369/3600 | 2.9913764947
07 |  1/49 |                       266681/176400 | 3.0117739478
08 |  1/64 |                      1077749/705600 | 3.0272978567
09 |  1/81 |                     9778141/6350400 | 3.0395075896
10 | 1/100 |                     1968329/1270080 | 3.0493616360
11 | 1/121 |                 239437889/153679680 | 3.0574815067
12 | 1/144 |                 240505109/153679680 | 3.0642878178
13 | 1/169 |             40799043101/25971865920 | 3.0700753719
14 | 1/196 |             40931552621/25971865920 | 3.0750569156
15 | 1/225 |           205234915681/129859329600 | 3.0793898260
16 | 1/256 |           822968714749/519437318400 | 3.0831930203
17 | 1/289 |     238357395880861/150117385017600 | 3.0865580258
18 | 1/324 |     238820721143261/150117385017600 | 3.0895564350
19 | 1/361 | 86364397717734821/54192375991353600 | 3.0922450523
20 | 1/400 | 17299975731542641/10838475198270720 | 3.0946695241