Euler and infinity What do people mean when they say that Euler treated infinity differently? I read in various books that, today, mathematicians would not approve of Euler's methods and his proofs lacked rigor. Can anyone elaborate?
Edit:
If I remember correctly Euler's original solution to the Basel problem is as follows.
Using Taylor series for $\sin (s)/s$ we write
$$\sin (s)/s = 1 - {s^2}/3! + {s^4}/5! -  \cdots $$
but $\sin (s)/s$ vanishes at $\pm \pi$, $\pm 2\pi$, etc. hence
$$\frac{{\sin s}}{s} = {\left( {1 - \frac{s}{\pi }} \right)}{\left( {1 + \frac{s}{\pi }} \right)}{\left( {1 - \frac{s}{{2\pi }}} \right)}{\left( {1 + \frac{s}{{2\pi }}} \right)}{\left( {1 - \frac{s}{{3\pi }}} \right)}{\left( {1 + \frac{s}{{3\pi }}} \right)} \cdots$$
or
$$\frac{{\sin s}}{s} = {\left( {1 - \frac{{{s^2}}}{{{1^2}\pi^2}}} \right)}{\left( {1 - \frac{{{s^2}}}{{{2^2}{\pi ^2}}}} \right)}{\left( {1 - \frac{{{s^2}}}{{{3^2}{\pi ^2}}}} \right)} \cdots$$
which is
$$\frac{{\sin s}}{s} = 1 - \frac{{{s^2}}}{{{\pi ^2}}}{\left( {\frac{1}{{{1^2}}} + \frac{1}{{{2^2}}} + \frac{1}{{{3^2}}} +  \cdots } \right)} +  \cdots.$$
Equating coefficients yields
$$\zeta (2) = \frac{{{\pi ^2}}}{6}.$$
But $\pm \pi$, $\pm 2\pi$, etc. are also roots of ${e^s}\sin (s)/s$, correct? So equating coefficients does not give ${\pi ^2}/6$.

There is another way Euler "lacked rigour" in nowadays terms. 
He used the idea of "something infinitesimally small" in his Introductio in analysin infinitorum (chapter 7, §115). He just gave this meaning to a variable and identified the term with its limit. So he would have said "$\frac{1}{\delta}=0$ for $\delta$ infinitely small". 
(This is something people use to do nowadays - at least when they aren't mathematicians.)
Clearly Euler didn't have the notions of mathematics from Cauchy, Weierstrass and so on. So it's kind of mean to say he lacked rigour. (By the way: I recommend reading (or at least browsing) the Introductio once - it is quite interesting to see how he develops all these equalities between trigonomic, rational and exponential functions.)
