Some new examples of summation of divergent series from the viewpoint of distributions

Let $\{a_1,a_2,\dots ,a_n,\dots \}$ be a sequence of complex numbers which has at most polynomial growth and satisfies an extra assumption. In this paper, inspired by a recent work of Sasane, we give an explanation of the sum $$a_1+2a_2+3a_3+\cdots +n{a}_n+\cdots ,$$ and more generally, for any $k\in \mathbb{N},$ the sum $$1^k{a}_1+2^k{a}_2+3^k{a}_3+\cdots +n^k{a}_n+\cdots ,$$ from the viewpoint of distributions. As applications, we explain the following summation formulas \begin{equation*} \begin{aligned} 1^{k}-2^{k}+3^{k}-\cdots&=-\frac{E_{k}(0)}{2}, \ 1^{k}+2^{k}+3^{k}+\cdots&=-\frac{B_{k+1}}{k+1}, \ ε^{1}1^{k}+ε^{2}2^{k}+ε^{3}3^{k}+\cdots&=-\frac{B_{k+1}(ε)}{k+1}, \end{aligned} \end{equation*} where $E_k(0)$, $B_k$ and $B_k(\epsilon )$ are the Euler polynomials at 0, the Bernoulli numbers and the Apostol–Bernoulli numbers, respectively.