This question was posed originally on MSE, I put it here because I didn't receive the answer(s) I wished to see.

Dear MO-Community,

When I was trying to find closed-form representations for odd zeta-values, I used $$ \Gamma(z) = \frac{e^{-\gamma \cdot z}}{z} \prod_{n=1}^{\infty} \Big( 1 + \frac{z}{n} \Big)^{-1} e^{\frac{z}{n}} $$ and rearranged it to $$ \frac{\Gamma(z)}{e^{-\gamma \cdot z}} = \prod_{n=1}^{\infty} \Big( 1 + \frac{z}{n} \Big)^{-1} e^{z/n}. $$ As we know that $$\prod_{n=1}^{\infty} e^{z/n} = e^{z + z/2 + z/3 + \cdots + z/n} = e^{\zeta(1) z},$$ we can state that $$\prod_{n=1}^{\infty} \Big( 1 + \frac{z}{n} \Big) = \frac{e^{z(\zeta(1) - \gamma)}}{z\Gamma(z)} \tag 1$$ I then stumbled upon the Wikipedia page of Ramanujan Summation (see the bottom of the page), which I used to set $\zeta(1) = \gamma$ (which was, admittedly, a rather dangerous move. Remarkably, things went well eventually. Please don't stop reading). The $z^3$ -coefficient of both sides can now be obtained. Consider \begin{align*} (1-ax)(1-bx) &= 1 - (a+b)x + abx^2\\\ &= 1-(a+b)x + (1/2)((a+b)^2-(a^2+b^2)) \end{align*} and \begin{align*}(1-ax)(1-bx)(1-cx) &= 1 - (a + b + c)x\\\ &\qquad + (1/2)\Bigl((a + b + c)^2 - (a^2 + b^2 + c^2)\Bigr)x^2\\\ &\qquad -(abc)x^3. \end{align*}

We can also set
\begin{align*}
(abc)x^3 &= (1/3)\Bigl((a^3 + b^3 + c^3) - (a + b + c)\Bigr)\\\
&\qquad + (1/2)(a + b + c)^3 -(a + b + c)(a^2 + b^2 + c^2).
\end{align*}
It can be proved by induction that the x^3 term of $(1-ax)(1-bx)\cdots(1-nx)$ is equal to
$$\begin{array} {lll}
\phantom+ (1/3) \Bigl((a^3 + b^3 + c^3 +\cdots + n^3) - (a + b + c + \cdots + n)^3\Big) \\
+ (1/2)\Big((a + b + c + \cdots + n)^3 \\
\qquad\qquad -(a + b + c + \cdots + n)(a^2 + b^2 + c^2 + \cdots + n^2)\Big).
\end{array} \tag 2 $$
On the right side of equation (1), the $z^3$-term can be found by looking at the $z^3$ term of the Taylor expansion series of $1/(z \Gamma(z))$, which is $(1/3)\zeta(3) + (1/2)\zeta(2) + (1/6)\gamma^3$. We then use (2) to obtain the equality
$$ (1/6)\gamma^3 - (1/2)\gamma \pi^2 - (1/6) \psi^{(2)}(1) = 1/3)\zeta(3) + (1/2)\zeta(2) + (1/6)\gamma^3$$
to find that
$$\zeta(3) = - (1/2) \psi^{(2)} (1),\tag 3$$
which is a true result that has been
known (*known* should be a hyperlink but it isn't for some reason) for quite a long time. The important thing here is that I used $\zeta(1) = \gamma$, which isn't really true. Ramanujan assigned a summation value to the harmonic series (again, see Ramanujan Summation), and apparently it can be used to verify results and perhaps to prove other conjectures/solve problems.

My first question is: Is this a legitimate way to prove (3) ?

Generalizing this question:

When and how are divergent series and their summation values used in mathematics?

What are the 'rules' when dealing with summed divergent series and using them to (try to) find new results?

Thanks,

Max

As I suspect someone (I was thinking of Qiaochu Yuan himself) will add this too, I will add this question for him/her, as it is somewhat related.