Forgive me if this is a ridiculous question, maybe I'm just blanking here. But I was wondering something about limits of a sum of sequences withotu the condition that both be converging. Here's what I mean: I've seen that

If $(a_n)_{n \in \mathbb{N}}$ and $(b_n)_{n \in \mathbb{N}}$ are convergent real sequences, then $$\lim_{n \to \infty} \left( a_n + b_n \right) = \left( \lim_{n \to \infty} a_n \right) + \left( \lim_{n \to \infty} b_n \right) . $$

But is it ok to say the same thing if only $a_n$ converges and b is possibly divergent? Example: $\lim_{n \to \infty} a_n = L$ , can you say $$\lim_{n \to \infty} \left( a_n + b_n \right) = L + \left( \lim_{n \to \infty} b_n \right) . $$ Or is this just nonsensical as $b_n$ doesnt have a limit?

  • $\begingroup$ Depends on your convention for $x=y$. If the two values are both undefined then you might accept as true that they are equal according to one convention. If you use the convention that both must be defined and equal, then it is false. $\endgroup$ – Somos Feb 7 at 3:26
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    $\begingroup$ Let $c_n=a_n+b_n$ then $b_n=c_n-a_n$ so $c_n$ cannot be convergent when $a_n$ convergent and $b_n$ divergent. $\endgroup$ – Minz Feb 7 at 3:28
  • $\begingroup$ Thank you! In the end, I think I might have worried for nothing. I'M still not sure, but for the exercise I was doing, my sequence maybe well defined and convergent after all. $\endgroup$ – Antoine Feb 7 at 3:34

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