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Let us consider the convergente series: $∑_{n=1}^{∞}a_{n}=S$ where $S$ is a real number. I know about Rearrangements and unconditional convergence (https://en.wikipedia.org/wiki/Absolute_convergence). But I have the following problem: Is it possible to writte the following:

$$∑_{n=1}^{∞}a_{n}=S⇒∑_{n=2}^{∞}a_{n}=S-a₁$$ without considering the absolute or the the unconditional convergence of the series.

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Yes.

Setting $s_n:=\sum_{k=1}^na_k$ the statement $\sum_{n=1}^{\infty}a_n=S$ is actually the same as $\lim_{n\to\infty}s_n=S$.

If that is the case and $t_n:=\sum_{k=2}^na_k=s_n-a_1$ then we find easily that $\lim_{n\to\infty}t_n=S-a_1$ or equivalently: $$\sum_{n=2}^{\infty}a_n=S-a_1$$

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