<s>[INST] How to sum $\frac1{1\cdot 2\cdot 3\cdot 4} + \frac4{3\cdot 4\cdot 5\cdot 6} + \frac9{5\cdot 6\cdot 7\cdot 8} + \cdots$ quickly? The Problem:
$$\frac1{1\cdot 2\cdot 3\cdot 4} + \frac4{3\cdot 4\cdot 5\cdot 6} + \frac9{5\cdot 6\cdot 7\cdot 8} + \frac{16}{7 \cdot 8 \cdot 9 \cdot 10} + \cdots$$
Any smarter way to solve this may be within a minute or two?

I am adding my solution,If you are a student please don't read-on I find this particular problem to some sort of interesting so just try it once :-)
I started off with trying to find the $T_n$ th term
$$T_n = \frac{n^2}{(2n-1)(2n)(2n+1)(2n+2)}$$
$$ = \frac1{24} \times \biggl( \frac1{2n-1} + \frac3{2n+1} - \frac4{2n+2} \biggr)$$ 
$$ = \frac1{24} \times \biggl[\biggl( \frac1{2n-1} + \frac1{2n+1}\biggr) + 4 \times \biggl(\frac1{2n+1} - \frac1{2n+2} \biggr) \biggr]$$   
After this this becomes really easy,
  $$ S_\infty =   \frac1{24} \times \biggl[( 1 - \frac13 + \frac13 -  \frac15 +  \frac15 + \cdots ) + 4 \times (\ln 2 - \frac12)\biggr]$$
$$ = \frac16 \log_e 2 - \frac1{24}$$
But as you can see this approach is a bit tedious and it took me some 30 minutes to reduce the $T_n$ to that form. Any other smart approaches?
 [/INST] Problems like these are usually amenable to partial fraction decompositions.
From my experience with exam problems like these, once we do the partial fractions (which can be done quite quickly as described below), either we can do some sort of a telescoping sum, or write the resulting as a combination of well known series. In some cases, you can quickly and mechanically apply known estimates for well known series and find the sum. Of course, each problem is different and you might have to get creative.
Let us try that approach to your problem.

Step1: Splitting into Partial Fractions
You can try the following which should give you the partial fraction decomposition pretty quickly.
Set
$$ \dfrac{n}{2(2n-1)(2n+1)(2n+2)} = \dfrac{A}{2n-1} + \dfrac{B}{2n+1} + \dfrac{C}{2n+2}$$
Multiplying by $\displaystyle 2n-1$ and setting $\displaystyle n=1/2$ gives
$\displaystyle A = \dfrac{1/2}{2 \times 2 \times 3} = \dfrac{1}{24}$.
Multiply the original by $\displaystyle 2n+1$ and set $\displaystyle n=-1/2$. This gives us $B$.
$\displaystyle B = \dfrac{-1/2}{2 \times -2 \times 1} = \dfrac{1}{8}$
Multiply the original by $\displaystyle 2n+2$ and set $\displaystyle n = -1$. This gives us
$\displaystyle C = \dfrac{-1}{ 2 \times -3 \times -1 } = \dfrac{-1}{6}$.
Thus
$$ \dfrac{n}{2(2n-1)(2n+1)(2n+2)} = \dfrac{1}{24}\left(\dfrac{1}{2n-1} + \dfrac{3}{2n+1} - \dfrac{4}{2n+2}\right)$$
Apparently this is due to Heaviside: http://en.wikipedia.org/wiki/Heaviside_cover-up_method

Step2: Summing the series
Now to find the sum, you can use the following, which does not require any clever algebraic manipulations and so can be done quite quickly.
$$H_n = \sum_{j=1}^{n} \dfrac{1}{j} = \log n + \gamma + \mathcal{O}\left(\dfrac{1}{n}\right)$$
This gives us
$$\sum_{j=1}^{n} \dfrac{1}{2j - 1} = H_{2n} - \dfrac{1}{2} H_n = \log 2 + \frac{\log n}{2} + \gamma/2 + \mathcal{O}\left(\dfrac{1}{n}\right)$$
Thus your sum to $n$ terms is
$$\dfrac{1}{24}\left(\log 2 + \dfrac{\log n}{2} + \gamma/2 + 3\log 2 + 3 \dfrac{\log (n+1)}{2} - 3 + 3 \gamma/2  - 2\log (n+1)  - 2\gamma +2\right) + \mathcal{O}\left(\dfrac{1}{n}\right)$$
Since $\log(n+1)  = \log n + \mathcal{O}\left(\dfrac{1}{n}\right)$ we get that the sum to n terms is 
$$\dfrac{1}{24}(4 \log 2  -1) + \mathcal{O}\left(\dfrac{1}{n}\right)$$
As $\displaystyle n \to \infty$,
the limit is 
$$ \dfrac{1}{24}(4 \log 2 - 1)$$
Note: As J.M points out in the comments below, a very general method similar to the above can be found in Abramowitz and Stegun's book:

The technique you used for summing the
  series is described in Abramowitz and
  Stegun. (Harmonic numbers and
  digamma functions are trivially
  related).

Note1: As Mike points, out, there is an updated version of the book available here: http://dlmf.nist.gov/
</s>