Nice proof! Although now I wonder just how different it really is from [this one][1]. In [Proofs from the Book][2] *that* technique is  to show that $e^2$ and with some adjustments, $e^4$ are irrational.

One can certainly take other sums where the proof you mention applies, although one has to be more careful than I initially thought:

 **Flawed conjecture** (see the comments for a counterexample) You could take any convergent infinite sum $\sum \frac{n_i}{d_i}$ where  $\gcd(n_i,d_i)=1$ , and for every integer $q \gt 1$ there is a $j$ so that $q$ divides $d_i$ for all $i \gt j$. 

Strengthen the conditions to  

 - $d_1 | d_2 | \cdots$
 - for every integer $q \gt 1$ there is an $i$ so that $q$ divides $d_i$
 - $0 \lt |\frac{n_i}{d_i}| \lt \frac{1}{d_{i-1}}$


and the proof does seem to go through.  A question is which such sums give interesting real numbers.  

$\cos(1)=1-1/2+1/24-1/720\cdots$ works although $\cos(1)=\frac{e^i+e^{-i}}{2}$ so it depends what counts as a trivial relation. 



  [1]: http://en.wikipedia.org/wiki/Proof_that_e_is_irrational
  [2]: http://www.amazon.com/dp/3540636986