Some people seriously object to the statement that it is forbidden to claim the truth of any theorem (that uses any form of arithmetic, no matter how simple).

I think this follows trivially from Gödel's incompleteness theorem. Claiming that such theorem would be true, amounts to creating a new theorem consisting of the original theorem and a statement about its consistency. This new theorem is automatically inconsistent:

"For any formal effectively generated theory T including basic arithmetical truths and also certain truths about formal provability, if T includes a statement of its own …

Wolfram Alpha sometimes gives answers that are incorrect as Mathematica also does, but in Mathematica this is usually accompanied by error messages. This is what I got in Mathematica 8:
$\sum _{n=1}^{\infty } \left(\sum _{m=1}^{\infty } \frac{1}{n \left(m^2+n^2\right)}\right)$
becomes:
$\sum _{n=1}^{\infty } \frac{\pi n \coth (\pi n)-1}{2 n^3}$
=1.9887084003563...

$\sum _{n=1}^{\infty } \left(\sum _{m=1}^{\infty } \frac{1}{n^2 \left(m^2+n^2\right)}\right)$
tends to:
$\frac{\pi ^4}{72}$
=1.35290404213...