Interesting Identity: $\lim_{m\to\infty}\lim_{n\to\infty}\prod_{k=1}^{\infty}\prod_{j=2}^{m}(1+\phi_{n})^{nk^{-j}\Phi_{j}}=e^{\gamma}$ Define the following identities:
$$
\phi_{n}:=\frac{4}{\pi}\int_{0}^{\infty}\frac{\coth(nx^{-1})-xn^{-1}}{n(1+x^{2})^{2}}\;dx \qquad
\text{and}\qquad 
\Phi_{n}:=\frac{\cos(n\pi)}{n}
$$
$\forall n\in\mathbb{N}$. I want to show that :
$$
\lim_{m\to\infty}\lim_{n\to\infty}\prod_{k=1}^{\infty}\prod_{j=2}^{m}(1+\phi_{n})^{ nk^{-j}\Phi_{j}}=e^{\gamma}
$$
Where $\gamma$ is the Euler - Mascheroni constant.
What I did is that I used the linearity of integrals for $\phi_{n}$
$$
\frac{\pi}{4}\phi_{n}=\int_{0}^{\infty}\frac{\coth\left(\frac{n}{x}\right)}{n(1+x^{2})^{2}}\;dx-\int_{0}^{\infty}\frac{x}{n^{2}(1+x^{2})^{2}}\;dx
$$
The first one I define it as $I_{1}$ and the second one I define it as $I_{2}$. Afterwards I defined:
$$
\lambda_{n}(x):=\frac{\coth\left(\frac{n}{x}\right)}{n(1+x^{2})^{2}}\leq\left|\frac{1}{n(1+x^{2})^{2}}\right|\leq\left|\frac{1}{2n(1+x^{2})}\right|=:V_{n}(x)
$$
so that:
$$
\lim_{n\to\infty}\int_{0}^{\infty}V_{n}(x)\;dx=\lim_{n\to\infty}\frac{\pi}{4n}=0
$$
Applying the Lebesgue Dominated Convergence Theorem I get:
$$
\lim_{n\to\infty}\int_{0}^{\infty}\lambda_{n}(x)\;dx=0
\tag 1$$
Now note that:
$$
I_{2}=\int_{0}^{\infty}\frac{x}{n^{2}(1+x^{2})}\;dx=\frac{1}{2n^{2}}\quad\implies\quad\lim_{n\to\infty}I_{2}=0
\tag 2$$
$(1)$ and $(2)$ verify that $\lim_{n\to\infty}(1+\phi_{n})^{n}\leq\lim_{n\to\infty}(1+I_{1}-\displaystyle I_{2})^{n}=\lim_{n\to\infty}\left(1+\frac{1}{n}-\frac{2}{n\pi^{2}}\right)^{n}=e$
Now all I have to evaluate $\displaystyle\exp\left(\lim_{m\to\infty}\sum_{k=1}^{\infty}\sum_{j=2}^{\infty}\frac{\cos(j\pi)}{k^{j}j}\right)$ but this seems a bit confusing and misleading when I try it which is what I am stuck on.
 A: The expression under the limit(s) is equal to ${a_n}^{b_m}$, where $$a_n=(1+\phi_n)^n,\qquad b_m=\sum_{k=1}^\infty\sum_{j=2}^m\frac{(-1)^j}{jk^j}.$$
Let $f(t)=\coth(1/t)-t$, then $n\phi_n=(4/\pi)\int_0^\infty(1+x^2)^{-2}f(x/n)\,dx$. Observe that $f$ is decreasing (to check, take the derivative and use $\sinh z>z$ for $z>0$). Also, $f(0)=1$ if understood as the limit.
Thus, $n\mapsto f(x/n)$ is increasing for each fixed $x$, so DCT (or even MCT) is applicable, and $\lim\limits_{n\to\infty}n\phi_n=(4/\pi)\int_0^\infty(1+x^2)^{-2}f(0)\,dx=1$, hence $\lim\limits_{n\to\infty}a_n=e$ as you know.
As for $b_m$, the corresponding $\sum_{k=\color{red}{2}}^\infty\sum_{j=2}^{\color{red}{\infty}}$ is absolutely convergent, so that we're free to interchange the summations, hence to take $m\to\infty$ directly under the outer sum: $$\lim_{m\to\infty}b_m=\sum_{k=1}^\infty\sum_{j=2}^\infty\frac{(-1/k)^j}{j}=\sum_{k=1}^\infty\left[\frac1k-\log\left(1+\frac1k\right)\right]\\=\lim_{n\to\infty}\sum_{k=1}^n[\ldots]=\lim_{n\to\infty}\left[\left(\sum_{k=1}^n\frac1k\right)-\log(n+1)\right].$$ As $\lim\limits_{n\to\infty}\big(\log(n+1)-\log n\big)=0$, we then have $\lim\limits_{m\to\infty}b_m=\gamma$ "almost by definition".
Finally, the given limit is $e^\gamma$ because $(a,b)\mapsto a^b$ is continuous at $(e,\gamma)$.
