Summation of geometric series with non-real exponent does not evaluate

[oscarbenjamin]

From SO:
https://stackoverflow.com/questions/67789161/converge-of-a-symbolic-infinite-series

It seems that summation is unable to recognise a geometric series if the coefficient of the index variable in the exponent is non-real:

In [42]: s = Sum(2**(-k)*exp(I*k), (k, 1, oo))

In [43]: s
Out[43]: 
  ∞           
 ___          
 ╲            
  ╲    -k  ⅈ⋅k
  ╱   2  ⋅ℯ   
 ╱            
 ‾‾‾          
k = 1         

In [44]: s.doit()
Out[44]: 
  ∞           
 ___          
 ╲            
  ╲    -k  ⅈ⋅k
  ╱   2  ⋅ℯ   
 ╱            
 ‾‾‾          
k = 1         

In [45]: s.rewrite(exp).powsimp(deep=True)
Out[45]: 
  ∞                   
 ___                  
 ╲                    
  ╲    -k⋅log(2) + ⅈ⋅k
  ╱   ℯ               
 ╱                    
 ‾‾‾                  
k = 1                 

In [46]: s.rewrite(exp).powsimp(deep=True).doit()
Out[46]: 
  ∞                   
 ___                  
 ╲                    
  ╲    -k⋅log(2) + ⅈ⋅k
  ╱   ℯ               
 ╱                    
 ‾‾‾                  
k = 1                 

In [47]: Sum(x**k, (k, 1, oo)).doit().subs(x, exp(I - log(2)))
Out[47]: 
     ⅈ    
    ℯ     
──────────
  ⎛     ⅈ⎞
  ⎜    ℯ ⎟
2⋅⎜1 - ──⎟
  ⎝    2 ⎠

In [48]: Sum(x**k, (k, 1, oo)).doit().subs(x, exp(I - log(2))).n()
Out[48]: 0.0283939952189359 + 0.592837620697943⋅ⅈ

In [49]: s.n()
Out[49]: 0.0283939952189359 + 0.592837620697943⋅ⅈ

A simpler example:

In [55]: Sum(2**(k*(I-1)), (k, 1, oo)).doit()
Out[55]: 
  ∞              
 ___             
 ╲               
  ╲    k⋅(-1 + ⅈ)
  ╱   2          
 ╱               
 ‾‾‾             
k = 1 

[Maelstrom6]

I could be completely wrong but maybe #20531 has something to do with it. It certainly deals with the same part of the code. However, I tried it on 1.7.1 (before this merge) and it wasn't able to complete it.

Here is the main section that I think needs to be adjusted (give or take 10 lines above and below):

sympy/sympy/concrete/summations.py

Lines 1183 to 1196 in d720d82

	# Here we first attempt powsimp on f for easier matching with the
	# exponential pattern, and attempt expansion on the exponent for easier
	# matching with the linear pattern.
	e = f.powsimp().match(c1 ** wexp)
	if e is not None:
	e_exp = e.pop(wexp).expand().match(c2*i + c3)
	if e_exp is not None:
	e.update(e_exp)
	p = (c1**c3).subs(e)
	q = (c1**c2).subs(e)
	r = p*(q**a - q**(b + 1))/(1 - q)
	l = p*(b - a + 1)
	return Piecewise((l, Eq(q, S.One)), (r, True))