What is $F_{k,\infty}$, i.e., $F$ distribution when the second degree of freedom approaches infinity? What is $F_{k,\infty}$, i.e., $F$ distribution when the second degree of freedom approaches infinity? I'm wondering if there is a known distribution(such as $\chi_k^2$) that it converges to.

The F-distribution is the distribution of the ratio of two independent scaled chi-squared random variables, so we have the general form $F_{k_1,k_2} \sim (k_2 \chi^2_{k_1})/ (k_1 \chi^2_{k_2})$.  As $k_2 \rightarrow \infty$ we have the limit $\chi^2_{k_2}/k_2 \rightarrow 1$, so if we apply Slutsky's theorem we see that the F-distribution converges to a scaled chi-squared distribution:
$$F_{k_1,k_2} \rightarrow \frac{\chi^2_{k_1}}{k_1}.$$
Note that the scaled chi-squared distribution is an extremely useful distribution in statistical analysis (so much so that it should probably have taken the place of the chi-squared distribution; see discussion here).

Direct Proof: The above heuristic explanation uses the properties of the Snedecor F distribution based on its derivation from chi-squared random variables.  However, it is simple to give a direct proof the above result by taking limits of the density kernel.  The density kernel for the Snedecor F distribution can be written as:
$$\begin{align}
\text{Snedecor-F}(x|k_1,k_2)
&\propto \frac{1}{x} \sqrt{\frac{k_1^{k_1} k_2^{k_2} x^{k_1}}{(k_1x+k_2)^{k_1+k_2}}} \\[6pt]
&\propto x^{k_1/2 - 1} \bigg( \frac{k_2}{k_1x+k_2} \bigg)^{(k_1 + k_2)/2} \\[6pt]
&= x^{k_1/2 - 1} \bigg( 1 + \frac{k_1x}{k_2} \bigg)^{-(k_1 + k_2)/2} \\[6pt]
&= x^{k_1/2 - 1} \bigg( 1 + \frac{k_1+k_2}{k_2} \cdot \frac{k_1x}{2} \cdot \frac{1}{(k_1+k_2)/2} \bigg)^{-(k_1 + k_2)/2}. \\[6pt]
\end{align}$$
Taking $k_2 \rightarrow \infty$ we then obtain the limit:
$$\begin{align}
\text{Snedecor-F}(x|k_1,k_2)
&\propto x^{k_1/2 - 1} \bigg( 1 + \frac{k_1+k_2}{k_2} \cdot \frac{k_1x}{2} \cdot \frac{1}{(k_1+k_2)/2} \bigg)^{-(k_1 + k_2)/2} \\[6pt]
&\rightarrow x^{k_1/2 - 1} \bigg( 1 + \frac{k_1x}{2} \cdot \frac{1}{(k_1+k_2)/2} \bigg)^{-(k_1 + k_2)/2} \\[6pt]
&\rightarrow  x^{k_1/2 - 1} \exp \bigg( - \frac{k_1 x}{2} \bigg) \\[12pt]
&\propto \text{ScaledChiSq}(x|k_1). \\[6pt]
\end{align}$$
As can be seen, in the limit the kernel of the Snedecor F density approaches the kernel of the scaled chi-squared density.  Since convergence of the density kernels implies convergence in distribution, this is sufficient to prove that the Snedecor F distribution converges to the scaled chi-squared distribution.  $\blacksquare$
