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I’m trying to prove that if ${a_n}$ is convergent, and ${b_n}$ is divergent, then ${a_n} + {b_n}$ is a divergent series.

A proof a friend told, but I don’t understand how can be correct is:

Assume ${a_n} + {b_n}$ convereges.

- If $\lim_{x \to \infty}f'(x)=+\infty$ then $\lim_{x \to \infty}(f(x)-f(x-1))=+\infty$ and $\lim_{x \to \infty}f(x)=+\infty$.
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Therefore, $\lim_{n\to{\infty}}({a_n} + {b_n}) = \lim_{n\to{\infty}}({a_n}) + \lim_{n\to{\infty}}({b_n}) $

Contradiction, since ${b_n}$ doesn’t converge to a limit.

Why can limit arithmetic rules be used in this case? since ${b_n}$ is not a convergent series, I didn’t think you can use it to express a convergent sequence’s limit, and show “as if” ${b_n}$ has a limit and contradict the assumption using it.

Thanks.

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Rearrange the calculation so that you’re working with convergent sequences. Assume that $\lim\limits_{n\to\infty}(a_n+b_n)=L$ and $\lim\limits_{n\to\infty}a_n=M$; then

$$\lim_{n\to\infty}b_n=\lim_{n\to\infty}\big((a_n+b_n)-a_n\big)=\lim_{n\to\infty}(a_n+b_n)-\lim_{n\to\infty}a_n=L-M\;,$$

contradicting the assumption that $\langle b_n:n\in\Bbb N\rangle$ is divergent.

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