Corollary

If $(x_n), (y_n)$ are two sequences of points of $G$ such that $x_n = y_n$ except for a finite number of indices, and if the series whose general term is $x_n$ converges, then so does the series whose general term is $y_n$.

For the series whose general term is $x_n - y_n$ has all its terms zero from a certain index onwards.

This corollary may be put in the form that we may alter arbitrarily a finite number of terms of a convergent series without it ceasing to be convergent. In particular, if $y_n = 0$ for $n < m$, and $y_n = x_n$ for $n \geq m$, we see that the series whose general term is $y_n$ converges if and only if the series whose general term is $x_n$ converges; its sum is denoted by $\sum_{n=m}^{\infty} x_n$ and is called the residue of index $m$ of the series $(x_n)$. Since

the residue of index $m$ of a convergent series tends to 0 as $m$ tends to $+\infty$.

If a sequence $(x_n)_{n \in I}$ has as index set an infinite subset $I$ of $\mathbf{N}$, and if $\varphi$ denotes the strictly order-preserving bijection of $\mathbf{N}$ onto $I$, then the series defined by the sequence $(x_{\varphi(n)})_{n \in \mathbf{N}}$ is called, by abuse of language, the series defined by the sequence $(x_n)_{n \in I}$; if this series is convergent, its sum is denoted by $\sum_{n \in I}^{\infty} x_n$. It is immediately verified that this series converges if and only if the series whose general term is $(z_n)$ converges, where $z_n = x_n$ if $n \in I$ and $z_n = 0$ if $n \in \mathbf{C} I$.

It is important to notice that if the series defined by a sequence $(x_n)_{n \in \mathbf{N}}$ converges, there can exist infinite subsets $I$ of $\mathbf{N}$ such that the series defined by the subsequence $(x_n)_{n \in I}$ does not converge (see Exercise 5 and Chapter IV, § 7).

Propositions 4 and 5 extend immediately to series, and we leave their formulation in this case to the reader.