|
|
from sympy.concrete.summations import Sum |
|
|
from sympy.core.add import Add |
|
|
from sympy.core.numbers import (I, Rational, oo, pi) |
|
|
from sympy.core.singleton import S |
|
|
from sympy.core.symbol import (Symbol, symbols) |
|
|
from sympy.functions.combinatorial.factorials import (binomial, factorial, subfactorial) |
|
|
from sympy.functions.combinatorial.numbers import (fibonacci, harmonic) |
|
|
from sympy.functions.elementary.exponential import (exp, log) |
|
|
from sympy.functions.elementary.miscellaneous import sqrt |
|
|
from sympy.functions.elementary.trigonometric import (cos, sin) |
|
|
from sympy.functions.special.gamma_functions import gamma |
|
|
from sympy.series.limitseq import limit_seq |
|
|
from sympy.series.limitseq import difference_delta as dd |
|
|
from sympy.testing.pytest import raises, XFAIL |
|
|
from sympy.calculus.accumulationbounds import AccumulationBounds |
|
|
|
|
|
n, m, k = symbols('n m k', integer=True) |
|
|
|
|
|
|
|
|
def test_difference_delta(): |
|
|
e = n*(n + 1) |
|
|
e2 = e * k |
|
|
|
|
|
assert dd(e) == 2*n + 2 |
|
|
assert dd(e2, n, 2) == k*(4*n + 6) |
|
|
|
|
|
raises(ValueError, lambda: dd(e2)) |
|
|
raises(ValueError, lambda: dd(e2, n, oo)) |
|
|
|
|
|
|
|
|
def test_difference_delta__Sum(): |
|
|
e = Sum(1/k, (k, 1, n)) |
|
|
assert dd(e, n) == 1/(n + 1) |
|
|
assert dd(e, n, 5) == Add(*[1/(i + n + 1) for i in range(5)]) |
|
|
|
|
|
e = Sum(1/k, (k, 1, 3*n)) |
|
|
assert dd(e, n) == Add(*[1/(i + 3*n + 1) for i in range(3)]) |
|
|
|
|
|
e = n * Sum(1/k, (k, 1, n)) |
|
|
assert dd(e, n) == 1 + Sum(1/k, (k, 1, n)) |
|
|
|
|
|
e = Sum(1/k, (k, 1, n), (m, 1, n)) |
|
|
assert dd(e, n) == harmonic(n) |
|
|
|
|
|
|
|
|
def test_difference_delta__Add(): |
|
|
e = n + n*(n + 1) |
|
|
assert dd(e, n) == 2*n + 3 |
|
|
assert dd(e, n, 2) == 4*n + 8 |
|
|
|
|
|
e = n + Sum(1/k, (k, 1, n)) |
|
|
assert dd(e, n) == 1 + 1/(n + 1) |
|
|
assert dd(e, n, 5) == 5 + Add(*[1/(i + n + 1) for i in range(5)]) |
|
|
|
|
|
|
|
|
def test_difference_delta__Pow(): |
|
|
e = 4**n |
|
|
assert dd(e, n) == 3*4**n |
|
|
assert dd(e, n, 2) == 15*4**n |
|
|
|
|
|
e = 4**(2*n) |
|
|
assert dd(e, n) == 15*4**(2*n) |
|
|
assert dd(e, n, 2) == 255*4**(2*n) |
|
|
|
|
|
e = n**4 |
|
|
assert dd(e, n) == (n + 1)**4 - n**4 |
|
|
|
|
|
e = n**n |
|
|
assert dd(e, n) == (n + 1)**(n + 1) - n**n |
|
|
|
|
|
|
|
|
def test_limit_seq(): |
|
|
e = binomial(2*n, n) / Sum(binomial(2*k, k), (k, 1, n)) |
|
|
assert limit_seq(e) == S(3) / 4 |
|
|
assert limit_seq(e, m) == e |
|
|
|
|
|
e = (5*n**3 + 3*n**2 + 4) / (3*n**3 + 4*n - 5) |
|
|
assert limit_seq(e, n) == S(5) / 3 |
|
|
|
|
|
e = (harmonic(n) * Sum(harmonic(k), (k, 1, n))) / (n * harmonic(2*n)**2) |
|
|
assert limit_seq(e, n) == 1 |
|
|
|
|
|
e = Sum(k**2 * Sum(2**m/m, (m, 1, k)), (k, 1, n)) / (2**n*n) |
|
|
assert limit_seq(e, n) == 4 |
|
|
|
|
|
e = (Sum(binomial(3*k, k) * binomial(5*k, k), (k, 1, n)) / |
|
|
(binomial(3*n, n) * binomial(5*n, n))) |
|
|
assert limit_seq(e, n) == S(84375) / 83351 |
|
|
|
|
|
e = Sum(harmonic(k)**2/k, (k, 1, 2*n)) / harmonic(n)**3 |
|
|
assert limit_seq(e, n) == S.One / 3 |
|
|
|
|
|
raises(ValueError, lambda: limit_seq(e * m)) |
|
|
|
|
|
|
|
|
def test_alternating_sign(): |
|
|
assert limit_seq((-1)**n/n**2, n) == 0 |
|
|
assert limit_seq((-2)**(n+1)/(n + 3**n), n) == 0 |
|
|
assert limit_seq((2*n + (-1)**n)/(n + 1), n) == 2 |
|
|
assert limit_seq(sin(pi*n), n) == 0 |
|
|
assert limit_seq(cos(2*pi*n), n) == 1 |
|
|
assert limit_seq((S.NegativeOne/5)**n, n) == 0 |
|
|
assert limit_seq((Rational(-1, 5))**n, n) == 0 |
|
|
assert limit_seq((I/3)**n, n) == 0 |
|
|
assert limit_seq(sqrt(n)*(I/2)**n, n) == 0 |
|
|
assert limit_seq(n**7*(I/3)**n, n) == 0 |
|
|
assert limit_seq(n/(n + 1) + (I/2)**n, n) == 1 |
|
|
|
|
|
|
|
|
def test_accum_bounds(): |
|
|
assert limit_seq((-1)**n, n) == AccumulationBounds(-1, 1) |
|
|
assert limit_seq(cos(pi*n), n) == AccumulationBounds(-1, 1) |
|
|
assert limit_seq(sin(pi*n/2)**2, n) == AccumulationBounds(0, 1) |
|
|
assert limit_seq(2*(-3)**n/(n + 3**n), n) == AccumulationBounds(-2, 2) |
|
|
assert limit_seq(3*n/(n + 1) + 2*(-1)**n, n) == AccumulationBounds(1, 5) |
|
|
|
|
|
|
|
|
def test_limitseq_sum(): |
|
|
from sympy.abc import x, y, z |
|
|
assert limit_seq(Sum(1/x, (x, 1, y)) - log(y), y) == S.EulerGamma |
|
|
assert limit_seq(Sum(1/x, (x, 1, y)) - 1/y, y) is S.Infinity |
|
|
assert (limit_seq(binomial(2*x, x) / Sum(binomial(2*y, y), (y, 1, x)), x) == |
|
|
S(3) / 4) |
|
|
assert (limit_seq(Sum(y**2 * Sum(2**z/z, (z, 1, y)), (y, 1, x)) / |
|
|
(2**x*x), x) == 4) |
|
|
|
|
|
|
|
|
def test_issue_9308(): |
|
|
assert limit_seq(subfactorial(n)/factorial(n), n) == exp(-1) |
|
|
|
|
|
|
|
|
def test_issue_10382(): |
|
|
n = Symbol('n', integer=True) |
|
|
assert limit_seq(fibonacci(n+1)/fibonacci(n), n).together() == S.GoldenRatio |
|
|
|
|
|
|
|
|
def test_issue_11672(): |
|
|
assert limit_seq(Rational(-1, 2)**n, n) == 0 |
|
|
|
|
|
|
|
|
def test_issue_14196(): |
|
|
k, n = symbols('k, n', positive=True) |
|
|
m = Symbol('m') |
|
|
assert limit_seq(Sum(m**k, (m, 1, n)).doit()/(n**(k + 1)), n) == 1/(k + 1) |
|
|
|
|
|
|
|
|
def test_issue_16735(): |
|
|
assert limit_seq(5**n/factorial(n), n) == 0 |
|
|
|
|
|
|
|
|
def test_issue_19868(): |
|
|
assert limit_seq(1/gamma(n + S.One/2), n) == 0 |
|
|
|
|
|
|
|
|
@XFAIL |
|
|
def test_limit_seq_fail(): |
|
|
|
|
|
e = (harmonic(n)**3 * Sum(1/harmonic(k), (k, 1, n)) / |
|
|
(n * Sum(harmonic(k)/k, (k, 1, n)))) |
|
|
assert limit_seq(e, n) == 2 |
|
|
|
|
|
|
|
|
e = (Sum(2**k * binomial(2*k, k) / k**2, (k, 1, n)) / |
|
|
(Sum(2**k/k*2, (k, 1, n)) * Sum(binomial(2*k, k), (k, 1, n)))) |
|
|
assert limit_seq(e, n) == S(3) / 7 |
|
|
|
|
|
|
|
|
e = n**3*Sum(2**k/k**2, (k, 1, n))**2 / (2**n * Sum(2**k/k, (k, 1, n))) |
|
|
assert limit_seq(e, n) == 2 |
|
|
|
|
|
e = (harmonic(n) * Sum(2**k/k, (k, 1, n)) / |
|
|
(n * Sum(2**k*harmonic(k)/k**2, (k, 1, n)))) |
|
|
assert limit_seq(e, n) == 1 |
|
|
|
|
|
e = (Sum(2**k*factorial(k) / k**2, (k, 1, 2*n)) / |
|
|
(Sum(4**k/k**2, (k, 1, n)) * Sum(factorial(k), (k, 1, 2*n)))) |
|
|
assert limit_seq(e, n) == S(3) / 16 |
|
|
|
