Average value of $\frac{x'A^2x}{x'A^3x}$ over surface of $n$-dimensional sphere Suppose $A$ is a diagonal matrix with eigenvalues $1,\frac{1}{2},\frac{1}{3},\ldots,\frac{1}{n}$ and $x$ is drawn from standard Gaussian in $n$ dimensions. Define $z_n$ as follows
$$z_n=E_{x\sim \mathcal{N}\left(0, I_n\right)}\left[\frac{x^T A^2 x}{x^T A^3 x}\right]$$
Is it possible to prove or disprove the following?
$$\lim_{n\to \infty} z_n = 2$$
This is a crosspost from math.SE where several people provided altnernative characterizations of $z_n$ but which don't quite settle the question.
Motivation: $z_n$ is the expected value of learning rate which maximizes loss decrease for a gradient descent step on a quadratic $A$ and random starting point. If the limit is 2, this would be a nice mathematical illustration behind the heuristic used in practice, "in high dimensions -- set learning rate as high as possible"
 A: According to this answer,
$$z_\infty:=\lim_n z_n=I:=\int_0^\infty F(s)G(s)\,ds,$$
where
$$F(s):=\prod_{k=1}^\infty\frac{1}{\sqrt{1+2s/k^3}},\quad G(s):=\sum_{k=1}^\infty\frac k{k^3+2s}.$$
Mathematica can express $F$ and $G$ in terms of functions built-in in Mathematica (and these expressions should be rather straightforward to verify), and then the Mathematica command NIntegrate numerically evaluates $z_\infty=I$ as $\approx1.99218$ -- close to $2$, but not $2$; see the image of the corresponding Mathematica notebook below.
Using the facts that (i) $F$ and $G$ are positive, decreasing, and convex, and hence $FG$ is so, and that (ii) Mathematica can find the values of all its built-in functions with any degree of accuracy, it is rather straightforward to show that, in fact,  $I<2$.



A: Using the uniform integrability of what is under your expectation sign, we have
$$z_\infty:=\lim_n z_n=E(Q_2/Q_3),$$
where
$$Q_p:=\sum_1^\infty\frac{Z_k^2}{k^p}$$
and the $Z_k$'s are independent standard normal random variables (r.v.'s).
The value of $z_\infty$ is unlikely to be exactly $2$.
(The uniform integrability follows, say, by Rosenthal's inequality for $Q_2-EQ_2$ and the inequality $E(1/Y^2)<\infty$ for any r.v. $Y$ with a gamma distribution with shape parameter $>2$.)
