Here is a fairly general solution for the infinite sum of a real geometric term in an arbitrary variable var
from var = n0
to Infinity
. There may be some expressions involving special functions that can be simplified to a geometric term that Simplify
might miss, and the use of PowerExpand
may be omitted if Complex
bases are to be used. It was included here to handle Sqrt
and other rational powers of expressions. Whether or not to expand over Plus
is left to the reader; omit the second part of the definition of gsum
if it seems undesirable.
ClearAll[gsum];
SetAttributes[gsum, Listable];
Module[{r, n},
gsum[term_, var_, n0_ : 0] /; FreeQ[r = Simplify[
PowerExpand[Divide @@ (term /. {{var -> n + 1}, {var -> n}})],
n \[Element] Integers], n] :=
With[{a = term /. var -> 0}, a/(1 - r)];
gsum[t1_ + t2_, var_, , n0_ : 0] := gsum[t1, var, n0] + gsum[t2, var, n0];
]
The OP's examples:
terms = {b (a/bc t)^r, a/b (b t)^r, a (c/d t)^r};
gsum[terms, r, 1]
(* {(a b t)/(bc (1 - (a t)/bc)), (a t)/(1 - b t), (a c t)/(d (1 - (c t)/d))} *)
Further examples. The last contains terms that do not form geometric sums. The expression gsum
returns unevaluated (after threading over Plus
).
gsum[{3^r (a + 2)^(2 r), a^(5 r + 3), a^r + b^r + c^r, Sqrt[a^r], r^2 + 2 b^r + c^r^2}, r]
(* {1/(1 - 3 (2 + a)^2), a^3/(1 - a^5),
1/(1 - a) + 1/(1 - b) + 1/(1 - c),
1/(1 - Sqrt[a]), 2/(1 - b) + gsum[c^r^2, r] + gsum[r^2, r]} *)