from sympy import (
    Abs, And, binomial, Catalan, cos, Derivative, E, Eq, exp, EulerGamma,
    factorial, Function, harmonic, I, Integral, KroneckerDelta, log,
    nan, Ne, Or, oo, pi, Piecewise, Product, product, Rational, S, simplify,
    sin, sqrt, Sum, summation, Symbol, symbols, sympify, zeta, gamma, Le,
    Indexed, Idx, IndexedBase, prod)
from sympy.abc import a, b, c, d, f, k, m, x, y, z
from sympy.concrete.summations import telescopic
from sympy.utilities.pytest import XFAIL, raises
from sympy import simplify
from sympy.matrices import Matrix
from sympy.core.mod import Mod
from sympy.core.compatibility import range
n = Symbol('n', integer=True)
def test_karr_convention():
    # Test the Karr summation convention that we want to hold.
    # See his paper "Summation in Finite Terms" for a detailed
    # reasoning why we really want exactly this definition.
    # The convention is described on page 309 and essentially
    # in section 1.4, definition 3:
    #
    # \sum_{m <= i < n} f(i) 'has the obvious meaning'   for m < n
    # \sum_{m <= i < n} f(i) = 0                         for m = n
    # \sum_{m <= i < n} f(i) = - \sum_{n <= i < m} f(i)  for m > n
    #
    # It is important to note that he defines all sums with
    # the upper limit being *exclusive*.
    # In contrast, sympy and the usual mathematical notation has:
    #
    # sum_{i = a}^b f(i) = f(a) + f(a+1) + ... + f(b-1) + f(b)
    #
    # with the upper limit *inclusive*. So translating between
    # the two we find that:
    #
    # \sum_{m <= i < n} f(i) = \sum_{i = m}^{n-1} f(i)
    #
    # where we intentionally used two different ways to typeset the
    # sum and its limits.
    i = Symbol("i", integer=True)
    k = Symbol("k", integer=True)
    j = Symbol("j", integer=True)
    # A simple example with a concrete summand and symbolic limits.
    # The normal sum: m = k and n = k + j and therefore m < n:
    m = k
    n = k + j
    a = m
    b = n - 1
    S1 = Sum(i**2, (i, a, b)).doit()
    # The reversed sum: m = k + j and n = k and therefore m > n:
    m = k + j
    n = k
    a = m
    b = n - 1
    S2 = Sum(i**2, (i, a, b)).doit()
    assert simplify(S1 + S2) == 0
    # Test the empty sum: m = k and n = k and therefore m = n:
    m = k
    n = k
    a = m
    b = n - 1
    Sz = Sum(i**2, (i, a, b)).doit()
    assert Sz == 0
    # Another example this time with an unspecified summand and
    # numeric limits. (We can not do both tests in the same example.)
    f = Function("f")
    # The normal sum with m < n:
    m = 2
    n = 11
    a = m
    b = n - 1
    S1 = Sum(f(i), (i, a, b)).doit()
    # The reversed sum with m > n:
    m = 11
    n = 2
    a = m
    b = n - 1
    S2 = Sum(f(i), (i, a, b)).doit()
    assert simplify(S1 + S2) == 0
    # Test the empty sum with m = n:
    m = 5
    n = 5
    a = m
    b = n - 1
    Sz = Sum(f(i), (i, a, b)).doit()
    assert Sz == 0
    e = Piecewise((exp(-i), Mod(i, 2) > 0), (0, True))
    s = Sum(e, (i, 0, 11))
    assert s.n(3) == s.doit().n(3)
def test_karr_proposition_2a():
    # Test Karr, page 309, proposition 2, part a
    i = Symbol("i", integer=True)
    u = Symbol("u", integer=True)
    v = Symbol("v", integer=True)
    def test_the_sum(m, n):
        # g
        g = i**3 + 2*i**2 - 3*i
        # f = Delta g
        f = simplify(g.subs(i, i+1) - g)
        # The sum
        a = m
        b = n - 1
        S = Sum(f, (i, a, b)).doit()
        # Test if Sum_{m <= i < n} f(i) = g(n) - g(m)
        assert simplify(S - (g.subs(i, n) - g.subs(i, m))) == 0
    # m < n
    test_the_sum(u,   u+v)
    # m = n
    test_the_sum(u,   u  )
    # m > n
    test_the_sum(u+v, u  )
def test_karr_proposition_2b():
    # Test Karr, page 309, proposition 2, part b
    i = Symbol("i", integer=True)
    u = Symbol("u", integer=True)
    v = Symbol("v", integer=True)
    w = Symbol("w", integer=True)
    def test_the_sum(l, n, m):
        # Summand
        s = i**3
        # First sum
        a = l
        b = n - 1
        S1 = Sum(s, (i, a, b)).doit()
        # Second sum
        a = l
        b = m - 1
        S2 = Sum(s, (i, a, b)).doit()
        # Third sum
        a = m
        b = n - 1
        S3 = Sum(s, (i, a, b)).doit()
        # Test if S1 = S2 + S3 as required
        assert S1 - (S2 + S3) == 0
    # l < m < n
    test_the_sum(u,     u+v,   u+v+w)
    # l < m = n
    test_the_sum(u,     u+v,   u+v  )
    # l < m > n
    test_the_sum(u,     u+v+w, v    )
    # l = m < n
    test_the_sum(u,     u,     u+v  )
    # l = m = n
    test_the_sum(u,     u,     u    )
    # l = m > n
    test_the_sum(u+v,   u+v,   u    )
    # l > m < n
    test_the_sum(u+v,   u,     u+w  )
    # l > m = n
    test_the_sum(u+v,   u,     u    )
    # l > m > n
    test_the_sum(u+v+w, u+v,   u    )
def test_arithmetic_sums():
    assert summation(1, (n, a, b)) == b - a + 1
    assert Sum(S.NaN, (n, a, b)) is S.NaN
    assert Sum(x, (n, a, a)).doit() == x
    assert Sum(x, (x, a, a)).doit() == a
    assert Sum(x, (n, 1, a)).doit() == a*x
    lo, hi = 1, 2
    s1 = Sum(n, (n, lo, hi))
    s2 = Sum(n, (n, hi, lo))
    assert s1 != s2
    assert s1.doit() == 3 and s2.doit() == 0
    lo, hi = x, x + 1
    s1 = Sum(n, (n, lo, hi))
    s2 = Sum(n, (n, hi, lo))
    assert s1 != s2
    assert s1.doit() == 2*x + 1 and s2.doit() == 0
    assert Sum(Integral(x, (x, 1, y)) + x, (x, 1, 2)).doit() == \
        y**2 + 2
    assert summation(1, (n, 1, 10)) == 10
    assert summation(2*n, (n, 0, 10**10)) == 100000000010000000000
    assert summation(4*n*m, (n, a, 1), (m, 1, d)).expand() == \
        2*d + 2*d**2 + a*d + a*d**2 - d*a**2 - a**2*d**2
    assert summation(cos(n), (n, -2, 1)) == cos(-2) + cos(-1) + cos(0) + cos(1)
    assert summation(cos(n), (n, x, x + 2)) == cos(x) + cos(x + 1) + cos(x + 2)
    assert isinstance(summation(cos(n), (n, x, x + S.Half)), Sum)
    assert summation(k, (k, 0, oo)) == oo
def test_polynomial_sums():
    assert summation(n**2, (n, 3, 8)) == 199
    assert summation(n, (n, a, b)) == \
        ((a + b)*(b - a + 1)/2).expand()
    assert summation(n**2, (n, 1, b)) == \
        ((2*b**3 + 3*b**2 + b)/6).expand()
    assert summation(n**3, (n, 1, b)) == \
        ((b**4 + 2*b**3 + b**2)/4).expand()
    assert summation(n**6, (n, 1, b)) == \
        ((6*b**7 + 21*b**6 + 21*b**5 - 7*b**3 + b)/42).expand()
def test_geometric_sums():
    assert summation(pi**n, (n, 0, b)) == (1 - pi**(b + 1)) / (1 - pi)
    assert summation(2 * 3**n, (n, 0, b)) == 3**(b + 1) - 1
    assert summation(Rational(1, 2)**n, (n, 1, oo)) == 1
    assert summation(2**n, (n, 0, b)) == 2**(b + 1) - 1
    assert summation(2**n, (n, 1, oo)) == oo
    assert summation(2**(-n), (n, 1, oo)) == 1
    assert summation(3**(-n), (n, 4, oo)) == Rational(1, 54)
    assert summation(2**(-4*n + 3), (n, 1, oo)) == Rational(8, 15)
    assert summation(2**(n + 1), (n, 1, b)).expand() == 4*(2**b - 1)
    # issue 6664:
    assert summation(x**n, (n, 0, oo)) == \
        Piecewise((1/(-x + 1), Abs(x) < 1), (Sum(x**n, (n, 0, oo)), True))
    assert summation(-2**n, (n, 0, oo)) == -oo
    assert summation(I**n, (n, 0, oo)) == Sum(I**n, (n, 0, oo))
    # issue 6802:
    assert summation((-1)**(2*x + 2), (x, 0, n)) == n + 1
    assert summation((-2)**(2*x + 2), (x, 0, n)) == 4*4**(n + 1)/S(3) - S(4)/3
    assert summation((-1)**x, (x, 0, n)) == -(-1)**(n + 1)/S(2) + S(1)/2
    assert summation(y**x, (x, a, b)) == \
        Piecewise((-a + b + 1, Eq(y, 1)), ((y**a - y**(b + 1))/(-y + 1), True))
    assert summation((-2)**(y*x + 2), (x, 0, n)) == \
        4*Piecewise((n + 1, Eq((-2)**y, 1)),
                    ((-(-2)**(y*(n + 1)) + 1)/(-(-2)**y + 1), True))
    # issue 8251:
    assert summation((1/(n + 1)**2)*n**2, (n, 0, oo)) == oo
    #issue 9908:
    assert Sum(1/(n**3 - 1), (n, -oo, -2)).doit() == summation(1/(n**3 - 1), (n, -oo, -2))
    #issue 11642:
    result = Sum(0.5**n, (n, 1, oo)).doit()
    assert result == 1
    assert result.is_Float
    result = Sum(0.25**n, (n, 1, oo)).doit()
    assert result == S(1)/3
    assert result.is_Float
    result = Sum(0.99999**n, (n, 1, oo)).doit()
    assert result == 99999
    assert result.is_Float
    result = Sum(Rational(1, 2)**n, (n, 1, oo)).doit()
    assert result == 1
    assert not result.is_Float
    result = Sum(Rational(3, 5)**n, (n, 1, oo)).doit()
    assert result == S(3)/2
    assert not result.is_Float
    assert Sum(1.0**n, (n, 1, oo)).doit() == oo
    assert Sum(2.43**n, (n, 1, oo)).doit() == oo
    # Issue 13979:
    i, k, q = symbols('i k q', integer=True)
    result = summation(
        exp(-2*I*pi*k*i/n) * exp(2*I*pi*q*i/n) / n, (i, 0, n - 1)
    )
    assert result.simplify() == Piecewise(
            (1, Eq(exp(2*I*pi*(-k + q)/n), 1)), (0, True)
    )
def test_harmonic_sums():
    assert summation(1/k, (k, 0, n)) == Sum(1/k, (k, 0, n))
    assert summation(1/k, (k, 1, n)) == harmonic(n)
    assert summation(n/k, (k, 1, n)) == n*harmonic(n)
    assert summation(1/k, (k, 5, n)) == harmonic(n) - harmonic(4)
def test_composite_sums():
    f = Rational(1, 2)*(7 - 6*n + Rational(1, 7)*n**3)
    s = summation(f, (n, a, b))
    assert not isinstance(s, Sum)
    A = 0
    for i in range(-3, 5):
        A += f.subs(n, i)
    B = s.subs(a, -3).subs(b, 4)
    assert A == B
def test_hypergeometric_sums():
    assert summation(
        binomial(2*k, k)/4**k, (k, 0, n)) == (1 + 2*n)*binomial(2*n, n)/4**n
def test_other_sums():
    f = m**2 + m*exp(m)
    g = 3*exp(S(3)/2)/2 + exp(S(1)/2)/2 - exp(-S(1)/2)/2 - 3*exp(-S(3)/2)/2 + 5
    assert summation(f, (m, -S(3)/2, S(3)/2)).expand() == g
    assert summation(f, (m, -1.5, 1.5)).evalf().epsilon_eq(g.evalf(), 1e-10)
fac = factorial
def NS(e, n=15, **options):
    return str(sympify(e).evalf(n, **options))
def test_evalf_fast_series():
    # Euler transformed series for sqrt(1+x)
    assert NS(Sum(
        fac(2*n + 1)/fac(n)**2/2**(3*n + 1), (n, 0, oo)), 100) == NS(sqrt(2), 100)
    # Some series for exp(1)
    estr = NS(E, 100)
    assert NS(Sum(1/fac(n), (n, 0, oo)), 100) == estr
    assert NS(1/Sum((1 - 2*n)/fac(2*n), (n, 0, oo)), 100) == estr
    assert NS(Sum((2*n + 1)/fac(2*n), (n, 0, oo)), 100) == estr
    assert NS(Sum((4*n + 3)/2**(2*n + 1)/fac(2*n + 1), (n, 0, oo))**2, 100) == estr
    pistr = NS(pi, 100)
    # Ramanujan series for pi
    assert NS(9801/sqrt(8)/Sum(fac(
        4*n)*(1103 + 26390*n)/fac(n)**4/396**(4*n), (n, 0, oo)), 100) == pistr
    assert NS(1/Sum(
        binomial(2*n, n)**3 * (42*n + 5)/2**(12*n + 4), (n, 0, oo)), 100) == pistr
    # Machin's formula for pi
    assert NS(16*Sum((-1)**n/(2*n + 1)/5**(2*n + 1), (n, 0, oo)) -
        4*Sum((-1)**n/(2*n + 1)/239**(2*n + 1), (n, 0, oo)), 100) == pistr
    # Apery's constant
    astr = NS(zeta(3), 100)
    P = 126392*n**5 + 412708*n**4 + 531578*n**3 + 336367*n**2 + 104000* \
        n + 12463
    assert NS(Sum((-1)**n * P / 24 * (fac(2*n + 1)*fac(2*n)*fac(
        n))**3 / fac(3*n + 2) / fac(4*n + 3)**3, (n, 0, oo)), 100) == astr
    assert NS(Sum((-1)**n * (205*n**2 + 250*n + 77)/64 * fac(n)**10 /
              fac(2*n + 1)**5, (n, 0, oo)), 100) == astr
def test_evalf_fast_series_issue_4021():
    # Catalan's constant
    assert NS(Sum((-1)**(n - 1)*2**(8*n)*(40*n**2 - 24*n + 3)*fac(2*n)**3*
        fac(n)**2/n**3/(2*n - 1)/fac(4*n)**2, (n, 1, oo))/64, 100) == \
        NS(Catalan, 100)
    astr = NS(zeta(3), 100)
    assert NS(5*Sum(
        (-1)**(n - 1)*fac(n)**2 / n**3 / fac(2*n), (n, 1, oo))/2, 100) == astr
    assert NS(Sum((-1)**(n - 1)*(56*n**2 - 32*n + 5) / (2*n - 1)**2 * fac(n - 1)
              **3 / fac(3*n), (n, 1, oo))/4, 100) == astr
def test_evalf_slow_series():
    assert NS(Sum((-1)**n / n, (n, 1, oo)), 15) == NS(-log(2), 15)
    assert NS(Sum((-1)**n / n, (n, 1, oo)), 50) == NS(-log(2), 50)
    assert NS(Sum(1/n**2, (n, 1, oo)), 15) == NS(pi**2/6, 15)
    assert NS(Sum(1/n**2, (n, 1, oo)), 100) == NS(pi**2/6, 100)
    assert NS(Sum(1/n**2, (n, 1, oo)), 500) == NS(pi**2/6, 500)
    assert NS(Sum((-1)**n / (2*n + 1)**3, (n, 0, oo)), 15) == NS(pi**3/32, 15)
    assert NS(Sum((-1)**n / (2*n + 1)**3, (n, 0, oo)), 50) == NS(pi**3/32, 50)
def test_euler_maclaurin():
    # Exact polynomial sums with E-M
    def check_exact(f, a, b, m, n):
        A = Sum(f, (k, a, b))
        s, e = A.euler_maclaurin(m, n)
        assert (e == 0) and (s.expand() == A.doit())
    check_exact(k**4, a, b, 0, 2)
    check_exact(k**4 + 2*k, a, b, 1, 2)
    check_exact(k**4 + k**2, a, b, 1, 5)
    check_exact(k**5, 2, 6, 1, 2)
    check_exact(k**5, 2, 6, 1, 3)
    assert Sum(x-1, (x, 0, 2)).euler_maclaurin(m=30, n=30, eps=2**-15) == (0, 0)
    # Not exact
    assert Sum(k**6, (k, a, b)).euler_maclaurin(0, 2)[1] != 0
    # Numerical test
    for m, n in [(2, 4), (2, 20), (10, 20), (18, 20)]:
        A = Sum(1/k**3, (k, 1, oo))
        s, e = A.euler_maclaurin(m, n)
        assert abs((s - zeta(3)).evalf()) < e.evalf()
def test_evalf_euler_maclaurin():
    assert NS(Sum(1/k**k, (k, 1, oo)), 15) == '1.29128599706266'
    assert NS(Sum(1/k**k, (k, 1, oo)),
              50) == '1.2912859970626635404072825905956005414986193682745'
    assert NS(Sum(1/k - log(1 + 1/k), (k, 1, oo)), 15) == NS(EulerGamma, 15)
    assert NS(Sum(1/k - log(1 + 1/k), (k, 1, oo)), 50) == NS(EulerGamma, 50)
    assert NS(Sum(log(k)/k**2, (k, 1, oo)), 15) == '0.937548254315844'
    assert NS(Sum(log(k)/k**2, (k, 1, oo)),
              50) == '0.93754825431584375370257409456786497789786028861483'
    assert NS(Sum(1/k, (k, 1000000, 2000000)), 15) == '0.693147930560008'
    assert NS(Sum(1/k, (k, 1000000, 2000000)),
              50) == '0.69314793056000780941723211364567656807940638436025'
def test_evalf_symbolic():
    f, g = symbols('f g', cls=Function)
    # issue 6328
    expr = Sum(f(x), (x, 1, 3)) + Sum(g(x), (x, 1, 3))
    assert expr.evalf() == expr
def test_evalf_issue_3273():
    assert Sum(0, (k, 1, oo)).evalf() == 0
def test_simple_products():
    assert Product(S.NaN, (x, 1, 3)) is S.NaN
    assert product(S.NaN, (x, 1, 3)) is S.NaN
    assert Product(x, (n, a, a)).doit() == x
    assert Product(x, (x, a, a)).doit() == a
    assert Product(x, (y, 1, a)).doit() == x**a
    lo, hi = 1, 2
    s1 = Product(n, (n, lo, hi))
    s2 = Product(n, (n, hi, lo))
    assert s1 != s2
    # This IS correct according to Karr product convention
    assert s1.doit() == 2
    assert s2.doit() == 1
    lo, hi = x, x + 1
    s1 = Product(n, (n, lo, hi))
    s2 = Product(n, (n, hi, lo))
    s3 = 1 / Product(n, (n, hi + 1, lo - 1))
    assert s1 != s2
    # This IS correct according to Karr product convention
    assert s1.doit() == x*(x + 1)
    assert s2.doit() == 1
    assert s3.doit() == x*(x + 1)
    assert Product(Integral(2*x, (x, 1, y)) + 2*x, (x, 1, 2)).doit() == \
        (y**2 + 1)*(y**2 + 3)
    assert product(2, (n, a, b)) == 2**(b - a + 1)
    assert product(n, (n, 1, b)) == factorial(b)
    assert product(n**3, (n, 1, b)) == factorial(b)**3
    assert product(3**(2 + n), (n, a, b)) \
        == 3**(2*(1 - a + b) + b/2 + (b**2)/2 + a/2 - (a**2)/2)
    assert product(cos(n), (n, 3, 5)) == cos(3)*cos(4)*cos(5)
    assert product(cos(n), (n, x, x + 2)) == cos(x)*cos(x + 1)*cos(x + 2)
    assert isinstance(product(cos(n), (n, x, x + S.Half)), Product)
    # If Product managed to evaluate this one, it most likely got it wrong!
    assert isinstance(Product(n**n, (n, 1, b)), Product)
def test_rational_products():
    assert simplify(product(1 + 1/n, (n, a, b))) == (1 + b)/a
    assert simplify(product(n + 1, (n, a, b))) == gamma(2 + b)/gamma(1 + a)
    assert simplify(product((n + 1)/(n - 1), (n, a, b))) == b*(1 + b)/(a*(a - 1))
    assert simplify(product(n/(n + 1)/(n + 2), (n, a, b))) == \
        a*gamma(a + 2)/(b + 1)/gamma(b + 3)
    assert simplify(product(n*(n + 1)/(n - 1)/(n - 2), (n, a, b))) == \
        b**2*(b - 1)*(1 + b)/(a - 1)**2/(a*(a - 2))
def test_wallis_product():
    # Wallis product, given in two different forms to ensure that Product
    # can factor simple rational expressions
    A = Product(4*n**2 / (4*n**2 - 1), (n, 1, b))
    B = Product((2*n)*(2*n)/(2*n - 1)/(2*n + 1), (n, 1, b))
    R = pi*gamma(b + 1)**2/(2*gamma(b + S(1)/2)*gamma(b + S(3)/2))
    assert simplify(A.doit()) == R
    assert simplify(B.doit()) == R
    # This one should eventually also be doable (Euler's product formula for sin)
    # assert Product(1+x/n**2, (n, 1, b)) == ...
def test_telescopic_sums():
    #checks also input 2 of comment 1 issue 4127
    assert Sum(1/k - 1/(k + 1), (k, 1, n)).doit() == 1 - 1/(1 + n)
    f = Function("f")
    assert Sum(
        f(k) - f(k + 2), (k, m, n)).doit() == -f(1 + n) - f(2 + n) + f(m) + f(1 + m)
    assert Sum(cos(k) - cos(k + 3), (k, 1, n)).doit() == -cos(1 + n) - \
        cos(2 + n) - cos(3 + n) + cos(1) + cos(2) + cos(3)
    # dummy variable shouldn't matter
    assert telescopic(1/m, -m/(1 + m), (m, n - 1, n)) == \
        telescopic(1/k, -k/(1 + k), (k, n - 1, n))
    assert Sum(1/x/(x - 1), (x, a, b)).doit() == -((a - b - 1)/(b*(a - 1)))
def test_sum_reconstruct():
    s = Sum(n**2, (n, -1, 1))
    assert s == Sum(*s.args)
    raises(ValueError, lambda: Sum(x, x))
    raises(ValueError, lambda: Sum(x, (x, 1)))
def test_limit_subs():
    for F in (Sum, Product, Integral):
        assert F(a*exp(a), (a, -2, 2)) == F(a*exp(a), (a, -b, b)).subs(b, 2)
        assert F(a, (a, F(b, (b, 1, 2)), 4)).subs(F(b, (b, 1, 2)), c) == \
            F(a, (a, c, 4))
        assert F(x, (x, 1, x + y)).subs(x, 1) == F(x, (x, 1, y + 1))
def test_function_subs():
    f = Function("f")
    S = Sum(x*f(y),(x,0,oo),(y,0,oo))
    assert S.subs(f(y),y) == Sum(x*y,(x,0,oo),(y,0,oo))
    assert S.subs(f(x),x) == S
    raises(ValueError, lambda: S.subs(f(y),x+y) )
    S = Sum(x*log(y),(x,0,oo),(y,0,oo))
    assert S.subs(log(y),y) == S
    f = Symbol('f')
    S = Sum(x*f(y),(x,0,oo),(y,0,oo))
    assert S.subs(f(y),y) == Sum(x*y,(x,0,oo),(y,0,oo))
def test_equality():
    # if this fails remove special handling below
    raises(ValueError, lambda: Sum(x, x))
    r = symbols('x', real=True)
    for F in (Sum, Product, Integral):
        try:
            assert F(x, x) != F(y, y)
            assert F(x, (x, 1, 2)) != F(x, x)
            assert F(x, (x, x)) != F(x, x)  # or else they print the same
            assert F(1, x) != F(1, y)
        except ValueError:
            pass
        assert F(a, (x, 1, 2)) != F(a, (x, 1, 3))
        assert F(a, (x, 1, 2)) != F(b, (x, 1, 2))
        assert F(x, (x, 1, 2)) != F(r, (r, 1, 2))
        assert F(1, (x, 1, x)) != F(1, (y, 1, x))
        assert F(1, (x, 1, x)) != F(1, (y, 1, y))
    # issue 5265
    assert Sum(x, (x, 1, x)).subs(x, a) == Sum(x, (x, 1, a))
def test_Sum_doit():
    assert Sum(n*Integral(a**2), (n, 0, 2)).doit() == a**3
    assert Sum(n*Integral(a**2), (n, 0, 2)).doit(deep=False) == \
        3*Integral(a**2)
    assert summation(n*Integral(a**2), (n, 0, 2)) == 3*Integral(a**2)
    # test nested sum evaluation
    s = Sum( Sum( Sum(2,(z,1,n+1)), (y,x+1,n)), (x,1,n))
    assert 0 == (s.doit() - n*(n+1)*(n-1)).factor()
    assert Sum(KroneckerDelta(m, n), (m, -oo, oo)).doit() == Piecewise((1, And(-oo < n, n < oo)), (0, True))
    assert Sum(x*KroneckerDelta(m, n), (m, -oo, oo)).doit() == Piecewise((x, And(-oo < n, n < oo)), (0, True))
    assert Sum(Sum(KroneckerDelta(m, n), (m, 1, 3)), (n, 1, 3)).doit() == 3
    assert Sum(Sum(KroneckerDelta(k, m), (m, 1, 3)), (n, 1, 3)).doit() == \
           3 * Piecewise((1, And(S(1) <= k, k <= 3)), (0, True))
    assert Sum(f(n) * Sum(KroneckerDelta(m, n), (m, 0, oo)), (n, 1, 3)).doit() == \
           f(1) + f(2) + f(3)
    assert Sum(f(n) * Sum(KroneckerDelta(m, n), (m, 0, oo)), (n, 1, oo)).doit() == \
           Sum(Piecewise((f(n), And(Le(0, n), n < oo)), (0, True)), (n, 1, oo))
    l = Symbol('l', integer=True, positive=True)
    assert Sum(f(l) * Sum(KroneckerDelta(m, l), (m, 0, oo)), (l, 1, oo)).doit() == \
           Sum(f(l), (l, 1, oo))
    # issue 2597
    nmax = symbols('N', integer=True, positive=True)
    pw = Piecewise((1, And(S(1) <= n, n <= nmax)), (0, True))
    assert Sum(pw, (n, 1, nmax)).doit() == Sum(pw, (n, 1, nmax))
    q, s = symbols('q, s')
    assert summation(1/n**(2*s), (n, 1, oo)) == Piecewise((zeta(2*s), 2*s > 1),
        (Sum(n**(-2*s), (n, 1, oo)), True))
    assert summation(1/(n+1)**s, (n, 0, oo)) == Piecewise((zeta(s), s > 1),
        (Sum((n + 1)**(-s), (n, 0, oo)), True))
    assert summation(1/(n+q)**s, (n, 0, oo)) == Piecewise(
        (zeta(s, q), And(q > 0, s > 1)),
        (Sum((n + q)**(-s), (n, 0, oo)), True))
    assert summation(1/(n+q)**s, (n, q, oo)) == Piecewise(
        (zeta(s, 2*q), And(2*q > 0, s > 1)),
        (Sum((n + q)**(-s), (n, q, oo)), True))
    assert summation(1/n**2, (n, 1, oo)) == zeta(2)
    assert summation(1/n**s, (n, 0, oo)) == Sum(n**(-s), (n, 0, oo))
def test_Product_doit():
    assert Product(n*Integral(a**2), (n, 1, 3)).doit() == 2 * a**9 / 9
    assert Product(n*Integral(a**2), (n, 1, 3)).doit(deep=False) == \
        6*Integral(a**2)**3
    assert product(n*Integral(a**2), (n, 1, 3)) == 6*Integral(a**2)**3
def test_Sum_interface():
    assert isinstance(Sum(0, (n, 0, 2)), Sum)
    assert Sum(nan, (n, 0, 2)) is nan
    assert Sum(nan, (n, 0, oo)) is nan
    assert Sum(0, (n, 0, 2)).doit() == 0
    assert isinstance(Sum(0, (n, 0, oo)), Sum)
    assert Sum(0, (n, 0, oo)).doit() == 0
    raises(ValueError, lambda: Sum(1))
    raises(ValueError, lambda: summation(1))
def test_eval_diff():
    assert Sum(x, (x, 1, 2)).diff(x) == 0
    assert Sum(x*y, (x, 1, 2)).diff(x) == 0
    assert Sum(x*y, (y, 1, 2)).diff(x) == Sum(y, (y, 1, 2))
    e = Sum(x*y, (x, 1, a))
    assert e.diff(a) == Derivative(e, a)
    assert Sum(x*y, (x, 1, 3), (a, 2, 5)).diff(y).doit() == \
        Sum(x*y, (x, 1, 3), (a, 2, 5)).doit().diff(y) == 24
def test_hypersum():
    from sympy import sin
    assert simplify(summation(x**n/fac(n), (n, 1, oo))) == -1 + exp(x)
    assert summation((-1)**n * x**(2*n) / fac(2*n), (n, 0, oo)) == cos(x)
    assert simplify(summation((-1)**n*x**(2*n + 1) /
        factorial(2*n + 1), (n, 3, oo))) == -x + sin(x) + x**3/6 - x**5/120
    assert summation(1/(n + 2)**3, (n, 1, oo)) == -S(9)/8 + zeta(3)
    assert summation(1/n**4, (n, 1, oo)) == pi**4/90
    s = summation(x**n*n, (n, -oo, 0))
    assert s.is_Piecewise
    assert s.args[0].args[0] == -1/(x*(1 - 1/x)**2)
    assert s.args[0].args[1] == (abs(1/x) < 1)
    m = Symbol('n', integer=True, positive=True)
    assert summation(binomial(m, k), (k, 0, m)) == 2**m
def test_issue_4170():
    assert summation(1/factorial(k), (k, 0, oo)) == E
def test_is_commutative():
    from sympy.physics.secondquant import NO, F, Fd
    m = Symbol('m', commutative=False)
    for f in (Sum, Product, Integral):
        assert f(z, (z, 1, 1)).is_commutative is True
        assert f(z*y, (z, 1, 6)).is_commutative is True
        assert f(m*x, (x, 1, 2)).is_commutative is False
        assert f(NO(Fd(x)*F(y))*z, (z, 1, 2)).is_commutative is False
def test_is_zero():
    for func in [Sum, Product]:
        assert func(0, (x, 1, 1)).is_zero is True
        assert func(x, (x, 1, 1)).is_zero is None
def test_is_number():
    # is number should not rely on evaluation or assumptions,
    # it should be equivalent to `not foo.free_symbols`
    assert Sum(1, (x, 1, 1)).is_number is True
    assert Sum(1, (x, 1, x)).is_number is False
    assert Sum(0, (x, y, z)).is_number is False
    assert Sum(x, (y, 1, 2)).is_number is False
    assert Sum(x, (y, 1, 1)).is_number is False
    assert Sum(x, (x, 1, 2)).is_number is True
    assert Sum(x*y, (x, 1, 2), (y, 1, 3)).is_number is True
    assert Product(2, (x, 1, 1)).is_number is True
    assert Product(2, (x, 1, y)).is_number is False
    assert Product(0, (x, y, z)).is_number is False
    assert Product(1, (x, y, z)).is_number is False
    assert Product(x, (y, 1, x)).is_number is False
    assert Product(x, (y, 1, 2)).is_number is False
    assert Product(x, (y, 1, 1)).is_number is False
    assert Product(x, (x, 1, 2)).is_number is True
def test_free_symbols():
    for func in [Sum, Product]:
        assert func(1, (x, 1, 2)).free_symbols == set()
        assert func(0, (x, 1, y)).free_symbols == {y}
        assert func(2, (x, 1, y)).free_symbols == {y}
        assert func(x, (x, 1, 2)).free_symbols == set()
        assert func(x, (x, 1, y)).free_symbols == {y}
        assert func(x, (y, 1, y)).free_symbols == {x, y}
        assert func(x, (y, 1, 2)).free_symbols == {x}
        assert func(x, (y, 1, 1)).free_symbols == {x}
        assert func(x, (y, 1, z)).free_symbols == {x, z}
        assert func(x, (x, 1, y), (y, 1, 2)).free_symbols == set()
        assert func(x, (x, 1, y), (y, 1, z)).free_symbols == {z}
        assert func(x, (x, 1, y), (y, 1, y)).free_symbols == {y}
        assert func(x, (y, 1, y), (y, 1, z)).free_symbols == {x, z}
    assert Sum(1, (x, 1, y)).free_symbols == {y}
    # free_symbols answers whether the object *as written* has free symbols,
    # not whether the evaluated expression has free symbols
    assert Product(1, (x, 1, y)).free_symbols == {y}
def test_conjugate_transpose():
    A, B = symbols("A B", commutative=False)
    p = Sum(A*B**n, (n, 1, 3))
    assert p.adjoint().doit() == p.doit().adjoint()
    assert p.conjugate().doit() == p.doit().conjugate()
    assert p.transpose().doit() == p.doit().transpose()
def test_issue_4171():
    assert summation(factorial(2*k + 1)/factorial(2*k), (k, 0, oo)) == oo
    assert summation(2*k + 1, (k, 0, oo)) == oo
def test_issue_6273():
    assert Sum(x, (x, 1, n)).n(2, subs={n: 1}) == 1
def test_issue_6274():
    assert Sum(x, (x, 1, 0)).doit() == 0
    assert NS(Sum(x, (x, 1, 0))) == '0'
    assert Sum(n, (n, 10, 5)).doit() == -30
    assert NS(Sum(n, (n, 10, 5))) == '-30.0000000000000'
def test_simplify():
    y, t, v = symbols('y, t, v')
    assert simplify(Sum(x*y, (x, n, m), (y, a, k)) + \
        Sum(y, (x, n, m), (y, a, k))) == Sum(y * (x + 1), (x, n, m), (y, a, k))
    assert simplify(Sum(x, (x, n, m)) + Sum(x, (x, m + 1, a))) == \
        Sum(x, (x, n, a))
    assert simplify(Sum(x, (x, k + 1, a)) + Sum(x, (x, n, k))) == \
        Sum(x, (x, n, a))
    assert simplify(Sum(x, (x, k + 1, a)) + Sum(x + 1, (x, n, k))) == \
        Sum(x, (x, n, a)) + Sum(1, (x, n, k))
    assert simplify(Sum(x, (x, 0, 3)) * 3 + 3 * Sum(x, (x, 4, 6)) + \
        4 * Sum(z, (z, 0, 1))) == 4*Sum(z, (z, 0, 1)) + 3*Sum(x, (x, 0, 6))
    assert simplify(3*Sum(x**2, (x, a, b)) + Sum(x, (x, a, b))) == \
        Sum(x*(3*x + 1), (x, a, b))
    assert simplify(Sum(x**3, (x, n, k)) * 3 + 3 * Sum(x, (x, n, k)) + \
        4 * y * Sum(z, (z, n, k))) + 1 == \
            4*y*Sum(z, (z, n, k)) + 3*Sum(x**3 + x, (x, n, k)) + 1
    assert simplify(Sum(x, (x, a, b)) + 1 + Sum(x, (x, b + 1, c))) == \
        1 + Sum(x, (x, a, c))
    assert simplify(Sum(x, (t, a, b)) + Sum(y, (t, a, b)) + \
        Sum(x, (t, b+1, c))) == x * Sum(1, (t, a, c)) + y * Sum(1, (t, a, b))
    assert simplify(Sum(x, (t, a, b)) + Sum(x, (t, b+1, c)) + \
        Sum(y, (t, a, b))) == x * Sum(1, (t, a, c)) + y * Sum(1, (t, a, b))
    assert simplify(Sum(x, (t, a, b)) + 2 * Sum(x, (t, b+1, c))) == \
        simplify(Sum(x, (t, a, b)) + Sum(x, (t, b+1, c)) + Sum(x, (t, b+1, c)))
    assert simplify(Sum(x, (x, a, b))*Sum(x**2, (x, a, b))) == \
        Sum(x, (x, a, b)) * Sum(x**2, (x, a, b))
    assert simplify(Sum(x, (t, a, b)) + Sum(y, (t, a, b)) + Sum(z, (t, a, b))) \
        == (x + y + z) * Sum(1, (t, a, b))          # issue 8596
    assert simplify(Sum(x, (t, a, b)) + Sum(y, (t, a, b)) + Sum(z, (t, a, b)) + \
        Sum(v, (t, a, b))) == (x + y + z + v) * Sum(1, (t, a, b))  # issue 8596
    assert simplify(Sum(x * y, (x, a, b)) / (3 * y)) == \
        (Sum(x, (x, a, b)) / 3)
    assert simplify(Sum(Function('f')(x) * y * z, (x, a, b)) / (y * z)) \
        == Sum(Function('f')(x), (x, a, b))
    assert simplify(Sum(c * x, (x, a, b)) - c * Sum(x, (x, a, b))) == 0
    assert simplify(c * (Sum(x, (x, a, b))  + y)) == c * (y + Sum(x, (x, a, b)))
    assert simplify(c * (Sum(x, (x, a, b)) + y * Sum(x, (x, a, b)))) == \
        c * (y + 1) * Sum(x, (x, a, b))
    assert simplify(Sum(Sum(c * x, (x, a, b)), (y, a, b))) == \
                c * Sum(x, (x, a, b), (y, a, b))
    assert simplify(Sum((3 + y) * Sum(c * x, (x, a, b)), (y, a, b))) == \
                c * Sum((3 + y), (y, a, b)) * Sum(x, (x, a, b))
    assert simplify(Sum((3 + t) * Sum(c * t, (x, a, b)), (y, a, b))) == \
                c*t*(t + 3)*Sum(1, (x, a, b))*Sum(1, (y, a, b))
    assert simplify(Sum(Sum(d * t, (x, a, b - 1)) + \
                Sum(d * t, (x, b, c)), (t, a, b))) == \
                    d * Sum(1, (x, a, c)) * Sum(t, (t, a, b))
def test_change_index():


Answer:     b, v = symbols('b, v', integer = True)