Probably trivial ratio of alternating sums 1/((a/b)^Log[c2/c1])^b

(*start*)
Clear[k]
c1 = 2;
c2 = 3;
a = 1/5;
b = 7;
 $MaxExtraPrecision = 10000
Sum[(-1)^(k - 1)*(a/b)^(Log[c1*k]*b), {k, 1, 10000}];
N[%, 30]
Sum[(-1)^(k - 1)*(a/b)^(Log[c2*k]*b), {k, 1, 10000}];
N[%, 30]
%%%/%
N[1/((a/b)^Log[c2/c1])^b, 30]
(*end*)