I'm stuck on the following exercise: "Let $\sum_{n=m}^\infty a_n$ be a series of real numbers, and let $k\geq 0$ be an integer. If one of the two series $\sum_{n=m}^\infty a_n$ and $\sum_{n=m+k}^\infty a_n$ are convergent, then the other one is also, and we have the identity $\sum_{n=m}^\infty a_n = \sum_{n=m}^{m+k-1} a_n + \sum_{n=m+k}^\infty a_n$."
Since in a previous exercise I showed that: "$(a_n)_{n=m}^\infty$ converges to $L$ iff $(a_n)_{n=m'\geq m}^\infty$" converges to $L$" (where $(a_n)_{n=m}^\infty$ is a sequence of real numbers) I could use this to say that the thesis of the exercise follows from this and the fact that by definition of convergence $\sum_{n=m}^\infty a_n$ converges $\Rightarrow$ $(S_N)_{N=m}^\infty$ converges to some limit $L$ as $N\to\infty$ and $L = \sum_{n=m}^\infty a_n$(where, for any integer $N\geq m$, $S_N$ is the Nth partial sum $S_N:=\sum_{n=m}^N a_n$) but this also implies that $\sum_{n=m}^\infty a_n = \sum_{n=m+k}^\infty a_n$ (since, by the exercise I cited above, $(S_N)_{N=m}^\infty \to L$ and $(S_N)_{N=m+k}^\infty \to L$) but this would contradict the fact that $\sum_{n=m}^\infty a_n = \sum_{n=m}^{m+k-1} a_n + \sum_{n=m+k}^\infty a_n$.
Could someone explain to me how to resolve this apparent paradox? (the text of this exercise says: $\sum_{n=m}^\infty a_n = \sum_{n=m}^{m+k-1} a_n + \sum_{n=m+k}^\infty a_n$ and the text of the previous exercise I cited above implies that $\sum_{n=m}^\infty a_n = \sum_{n=m+k}^\infty a_n$).
Best regards,
lorenzo.
You are misunderstanding the relationship between a sequence and a series. You are correct in that $S_N := \sum_{n=m}^N a_n$, and $(S_N)_{N=m}^{\infty} \rightarrow L \Rightarrow \sum_{n=m}^\infty a_n = L$. The issue is that you claim that $(S_N)_{N=m+k}^{\infty} \rightarrow L \Rightarrow \sum_{n=m+k}^\infty a_n = L$, which is not true.
I think the source of your confusion is that the two implications are identical with $m$ replaced by $m+k$. The problem is that you defined $S_N$ in terms of $m$, so changing $m$ to $m+k$ means that you have to change $S_N$ to something else.
For a simpler example, let $F(N) := mN$. We have that $N=m \Rightarrow F(N) = m^2$. However, it's not true that $N=m+k \Rightarrow F(N) = (m+k)^2$. Simply swapping out a variable doesn't work because the notation hides the dependence. If we were to say $F(N,m) := mN$, then we could swap out $m$ with $m+k$ and have no problems, because the dependence of $F$ on $m$ is clearly stated.