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Bug introduced in 9.0 or earlier and fixed in 11.0.0
This infinite product is correct:
Product[1/(1 - 1/2^n), {n, 1, Infinity}]
(* 1/QPochhammer[1/2, 1/2] *)
Numerical value of this result is not equal to zero
N[1/QPochhammer[1/2, 1/2]]
(* 3.462746619455064 *)
But from following equivalent expression I get a wrong output:
Product[1/(1 - 1/2^(n-1)), {n, 2, Infinity}]
(* 0 *)
$Version
"10.3.1 for Mac OS X x86 (64-bit) (December 9, 2015)"
p = Product[1/(1 - 1/2^n), {n, 1, Infinity}]
(* 1/QPochhammer[1/2, 1/2] *)
Error:
Product[1/(1 - 1/2^(n - 1)), {n, 2, Infinity}]
(* 0/QPochhammer[2, 1/2] *)
Workarounds:
Product[1/(1 - 1/2^(n - 1)), {n, m, Infinity}] /. m -> 2
(* 1/QPochhammer[1/2, 1/2] *)
Limit[Product[1/(1 - 1/2^(n - 1)), {n, m, Infinity}], m -> 2]
(* 1/QPochhammer[1/2, 1/2] *)
Product[1/(1 - 1/2^(n - 1)), {n, 2, Infinity}, Regularization -> "Dirichlet"]
(* 1/QPochhammer[1/2, 1/2] *)
1/(2 QPochhammer[1/2, 1/2]), as it should be. $\endgroup$ – bbgodfrey Dec 19 '15 at 17:1810.0 for Microsoft Windows (64-bit)with Windows 8.1 $\endgroup$ – mattiav27 Dec 19 '15 at 17:210/QPochhammer[2, 1/2]on 10.3.1. $\endgroup$ – b.gates.you.know.what Dec 19 '15 at 17:24Infinityto some large integer instead, it evaluates correctly. $\endgroup$ – march Dec 19 '15 at 17:48