If $U$ is the limit of a sequence $$u_1,\,u_2,\,u_3,\,\ldots$$ of real or complex numbers, then $U$ can be expressed as the series sum $$U = u_1+\sum_{i=1}^\infty(u_{i+1}-u_i).$$

Proof. Let $\displaystyle s_n := u_1+\sum_{i=1}^{n-1}(u_{i+1}-u_i)$ We see that $$s_n = u_1+\sum_{i=1}^{n-1}u_{i+1}-\sum_{i=1}^{n-1}u_i = u_1+\sum_{j=2}^nu_j-\sum_{i=1}^{n-1}u_i = u_n$$ for all $n = 1,\,2,\,3,\,\ldots$ , Thus $$u_1+\sum_{i=1}^\infty(u_{i+1}-u_i) = \lim_{n\to\infty}s_n = \lim_{n\to\infty}u_n = U,$$ Q.E.D.