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Series $a_n$ is convergent and $b_n$ is not-convergent. Will the sum $a_n + b_n$ converge? I think it will not converge, But how do I show it?

I believe I have to use the definition.

$|a_n - A| < \epsilon$

$|b_n - B| >= \epsilon$

Then choose $N > n$ for both, and try to achieve and equation that shows $ | a_n + b_n - A - B | >= \epsilon $

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It is not converge. because $$b_n=(b_n+a_n)-a_n,$$ so if $\sum b_n+a_n $ converges then $b_n$ so is.

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No, since $$b_n=(a_n+b_n)-a_n.$$

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Hints:

By Contradiction, say $\sum ( a_n + b_n) $ converges, and write

$$ - \sum a_n + \sum (a_n + b_n) = \sum b_n $$

What happens now?

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