This is
a generalization
of my answer to
$\sum_{1\leq l \lt m\lt n} \dfrac{1}{5^l3^m2^n}$
I showed there that,
if $0 < a, b ,c < 1$,
$\begin{array}\\
s(a, b, c)
&=\sum_{1\leq l \lt m\lt n} a^lb^mc^n\\
&=\sum_{n=1}^{\infty}\sum_{m=1}^{n-1}\sum_{l=1}^{m-1}a^lb^mc^n\\
&=\dfrac{ab^2c^3}{(1-c)(1-bc)...
If somebody thinks that tag like this should be created, probably the way to go is to discuss it first on meta, the Tag management thread seems to be a good place for that.
@MartinSleziak Only now I have noticed that this time it was Marty Cohen again who created the tag.
I have noticed this was second time you have created (generalization) tag. (And in the past this tag was created by other users and removed each time.) I think that probably it would be better to discuss the tag on meta first, since it is unclear whether or not this tag can be useful. (Relevant discussion on meta: Should every new tag be discussed on meta before creation?) After you see this comment, feel free to ping me here or in tagging chatroom - then I can remove it. — Martin Sleziak57 secs ago
This is
a generalization
of my answer to
$\sum_{1\leq l \lt m\lt n} \dfrac{1}{5^l3^m2^n}$
I showed there that,
if $0 < a, b ,c < 1$,
$\begin{array}\\
s(a, b, c)
&=\sum_{1\leq l \lt m\lt n} a^lb^mc^n\\
&=\sum_{n=1}^{\infty}\sum_{m=1}^{n-1}\sum_{l=1}^{m-1}a^lb^mc^n\\
&=\dfrac{ab^2c^3}{(1-c)(1-bc)...
