Alternating prime series

Since $p_n=\Theta(n\log n)$ by the weak version of the PNT, we may compute the wanted limit through a convolution with an approximate identity:

$$ \lim_{x\to 0^+} \sum_{n\geq 1}p_n(x-1)^{n-1} = \lim_{n\to +\infty} \sum_{n\geq 1}p_n \int_{0}^{+\infty}n(x-1)^{n-1} e^{-nx}\,dx $$ where $$ \int_{1}^{+\infty}n(x-1)^{n-1}e^{-nx}\,dx = \frac{n!}{e^n n^n}\approx\frac{\sqrt{2\pi n}}{e^{2n}}$$ gives no issues, but $$ \int_{0}^{1}n(x-1)^{n-1} e^{-nx}\,dx=(-1)^{n-1}\int_{0}^{n}\left(1-\frac{x} {n}\right)^{n-1}e^{-x}\,dx $$ behaves like $\frac{(-1)^n}{2}$ for large values of $n$. In particular the existence of the wanted limit depends on the Cesàro summability of the sequence $\{(-1)^n p_n\}_{n\geq 1}$, hence on the distribution of prime gaps.

The result of Ping Ngai Chung and Shiyu Li mentioned on MO implies that the wanted limit does not exist.