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[GSoC] Concrete: Implemented Raabe's Test #18656

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merged 9 commits into from
Aug 17, 2020
Merged

[GSoC] Concrete: Implemented Raabe's Test #18656

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c02ece9
Implemented Raabe's Test
sachin-4099 Feb 14, 2020
d5c27a4
Added test and some simplifications to get rid of RecursionError
sachin-4099 Feb 15, 2020
553f35a
Made a bit fast
sachin-4099 Feb 15, 2020
5313914
Added more testcases and made some modifications
sachin-4099 Feb 16, 2020
32bd957
Made some modifications to testcases
sachin-4099 Feb 16, 2020
7c863b8
Merge branch 'master' into raabe
sachin-4099 Feb 16, 2020
91a4b4d
New testcase
sachin-4099 Feb 16, 2020
e1dd2bd
Merge branch 'master' into raabe
sachin-4099 Aug 16, 2020
5de4028
Made some modifications and added a testcase
sachin-4099 Aug 16, 2020
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29 changes: 23 additions & 6 deletions sympy/concrete/summations.py
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Original file line number Diff line number Diff line change
Expand Up @@ -421,7 +421,8 @@ def is_convergent(self):
sym = self.limits[0][0]
lower_limit = self.limits[0][1]
upper_limit = self.limits[0][2]
sequence_term = self.function
sequence_term = simplify(self.function)
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if len(sequence_term.free_symbols) > 1:
raise NotImplementedError("convergence checking for more than one symbol "
Expand All @@ -446,6 +447,7 @@ def is_convergent(self):

interval = Interval(lower_limit, upper_limit)


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# Piecewise function handle
if sequence_term.is_Piecewise:
for func, cond in sequence_term.args:
Expand All @@ -457,12 +459,13 @@ def is_convergent(self):

### -------- Divergence test ----------- ###
try:
lim_val = limit_seq(sequence_term, sym)
if lim_val is not None and lim_val.is_zero is False:
return S.false
lim_val = limit_seq(sequence_term, sym)
if lim_val is not None and lim_val.is_zero is False:
return S.false
except NotImplementedError:
pass


try:
lim_val_abs = limit_seq(abs(sequence_term), sym)
if lim_val_abs is not None and lim_val_abs.is_zero is False:
Expand Down Expand Up @@ -510,7 +513,22 @@ def is_convergent(self):
if abs(lim_ratio) < 1:
return S.true
except NotImplementedError:
pass
lim_ratio = None

### ---------- Raabe's test -------------- ###
if lim_ratio == 1: # ratio test inconclusive
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There was a discussion in the other pull request about whether or not this is needed.

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imo it is necessary...

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Why is it necessary?

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Are you talking about raabe's test or the condition??

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The condition. Your answer above says "imo it is necessary". I am sure that I agree but I would prefer it if you write an explicit reason (an opinion is not a reason).

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We implement Raabe's test only after ratio test fails, and the condition when ratio test fails is when it is 1, so definitely it is necessary, otherwise how will we come to know when to implement raabe's test.

test_val = sym*(sequence_term/
sequence_term.subs(sym, sym + 1) - 1)
test_val = test_val.gammasimp()
try:
lim_val = limit_seq(test_val, sym)
if lim_val is not None and lim_val.is_number:
if lim_val > 1:
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I think we can check (lim_val - 1).is_positive here, to allow it to potentially work for symbols with the proper assumptions.

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OK will change the line if lim_value > 1:

return S.true
if lim_val < 1:
return S.false
except NotImplementedError:
pass

### ----------- root test ---------------- ###
# lim = Limit(abs(sequence_term)**(1/sym), sym, S.Infinity)
Expand All @@ -529,7 +547,6 @@ def is_convergent(self):
if not dict_val[p].has(sym) and is_decreasing(dict_val[q], interval):
return S.true


### ------------- integral test -------------- ###
check_interval = None
maxima = solveset(sequence_term.diff(sym), sym, interval)
Expand Down
7 changes: 7 additions & 0 deletions sympy/concrete/tests/test_sums_products.py
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Original file line number Diff line number Diff line change
Expand Up @@ -8,6 +8,7 @@
from sympy.abc import a, b, c, d, k, m, x, y, z
from sympy.concrete.summations import telescopic, _dummy_with_inherited_properties_concrete
from sympy.concrete.expr_with_intlimits import ReorderError
from sympy.functions.combinatorial.factorials import rf
from sympy.testing.pytest import XFAIL, raises, slow
from sympy.matrices import \
Matrix, SparseMatrix, ImmutableDenseMatrix, ImmutableSparseMatrix
Expand Down Expand Up @@ -995,6 +996,9 @@ def test_is_convergent():
assert Sum((-1)**n, (n, 1, oo)).is_convergent() is S.false
assert Sum(log(1/n), (n, 2, oo)).is_convergent() is S.false

# Raabe's test --
assert Sum(Product((3*m),(m,1,n))/Product((3*m+4),(m,1,n)),(n,1,oo)).is_convergent() is S.true

# root test --
assert Sum((-12)**n/n, (n, 1, oo)).is_convergent() is S.false

Expand All @@ -1005,6 +1009,9 @@ def test_is_convergent():
assert Sum(1/n**Rational(6, 5), (n, 1, oo)).is_convergent() is S.true
assert Sum(2/(n*sqrt(n - 1)), (n, 2, oo)).is_convergent() is S.true
assert Sum(1/(sqrt(n)*sqrt(n)), (n, 2, oo)).is_convergent() is S.false
assert Sum(factorial(n) / factorial(n+2), (n, 1, oo)).is_convergent() is S.true
assert Sum(rf(5,n)/rf(7,n),(n,1,oo)).is_convergent() is S.true
assert Sum((rf(1, n)*rf(2, n))/(rf(3, n)*factorial(n)),(n,1,oo)).is_convergent() is S.false

# comparison test --
assert Sum(1/(n + log(n)), (n, 1, oo)).is_convergent() is S.false
Expand Down