sum (1+1/2=1/3+..1/m)/((m+q)(m+r)) (improved)

From: Leroy Quet (qqquet_at_mindspring.com)
Date: 09/22/04 

Date: 22 Sep 2004 14:11:43 -0700

I am reposting this message with a typo fixed and ascii-art added. 

----
Here is a simple result I might have very well posted before.
If so, I apologize.
Let H(m) = sum{k=1 to m} 1/k, the m-th harmonic number.
Let q and r be positive integers where q does not equal r.
Then:
sum{m=1 to oo} H(m) /((m+q)(m+r))
=
(G(r-1) - G(q-1) + (H(r-1))^2 - (H(q-1))^2)/(2 (r-q)),
where G(m) = sum{k=1 to m} 1/k^2.
In ascii-art:
oo
---
\      H(m)
/  -----------
--- (m+q)(m+r)
m=1
=
                          2         2
G(r-1) - G(q-1) + (H(r-1)) - (H(q-1))
--------------------------------------
             2 (r-q)
thanks,
Leroy Quet