concrete: fix finite geometric sums

Added powsimp().cancel() heuristics to catch some cases
when the term is not in a canonical form.

Added regression test.

Closes sympy/sympy#21557

diofant/concrete/summations.py

@@ -654,13 +654,13 @@ def eval_sum_symbolic(f, limits):
                 else:
                     return harmonic(b, abs(n)) - harmonic(a - 1, abs(n))
 
-    if not (a.has(oo, -oo) or
-            b.has(oo, -oo)):
+    if not (a.has(oo, -oo) or b.has(oo, -oo)):
         # Geometric terms
         c1 = Wild('c1', exclude=[i])
         c2 = Wild('c2', exclude=[i])
         c3 = Wild('c3', exclude=[i])
 
+        f = f.powsimp().cancel()
         e = f.match(c1**(c2*i + c3))
 
         if e is not None:

diofant/tests/concrete/test_sums_products.py

@@ -911,3 +911,12 @@ def test_sympyissue_10156():
 def test_sympyissue_15943():
     s = Sum(binomial(n, k)*factorial(n - k), (k, 0, n))
     assert s.doit().simplify() == E*(gamma(n + 1) - lowergamma(n + 1, 1))
+
+
+def test_sympyissue_21557():
+    f1 = 2**(-k)*exp(I*k)
+    assert summation(f1, (k, 1, oo)).simplify() == E**I/(2 - E**I)
+    f2 = 2**(k*(I - 1))
+    assert summation(f2, (k, 1, oo)).simplify() == 2**I/(2 - 2**I)
+    assert (summation(f2, (k, 1, x)).simplify() ==
+            2**(-x + I)*(-2**x + 2**(I*x))/(-2 + 2**I))

docs/release/notes-0.13.rst

@@ -88,3 +88,4 @@ These Sympy issues also were addressed:
 * :sympyissue:`21530`: Incorrect limit
 * :sympyissue:`21166`: Wrong integration result involving square root of absolute value
 * :sympyissue:`21549`: Bug: integrate(x*sqrt(abs(x)),(x,-1,0)) returns wrong result
+* :sympyissue:`21557`: Summation of geometric series with non-real exponent does not evaluate