What are the conditions for this equation to be valid

$\displaystyle\sum_{n=1}^{\infty} a_n + \displaystyle\sum_{n=1}^{\infty} b_n=\displaystyle\sum_{n=1}^{\infty}(a_n+ b_n)$ ?

In other words, when can we say that the left-hand side is equal to the right-hand side?

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    $\begingroup$ As long as two of the sums converge. $\endgroup$ – Angina Seng Aug 27 '17 at 12:52
  • $\begingroup$ @LordSharktheUnknown we already said that to our professor, but he's not satisfied. He needs a broader explanation and we couldn't figure it our. $\endgroup$ – Crunchy Aug 27 '17 at 12:55

If $$\sum_{n=1}^{\infty} a_n =s \hspace{4mm}\mbox{ and }\hspace{4mm}\sum_{n=1}^{\infty} b_n=t$$ with $s,t\in \mathbb{R}$, then $$\sum_{n=1}^{\infty} (a_n+b_n) = s+t. $$ Furthermore, $$ \sum_{n=1}^{\infty}(ka_n) = ks $$ for every $k\in \mathbb{R}$.

$\textbf{Theorem}$. (Cauchy criterion for series) The infinite series $\sum_n a_n$ converges if and only if for each $\varepsilon>0$, there exists $N\in \mathbb{N}$ such that if $n\geq m\geq N$, then $$ |a_m+a_{m+1}+\ldots + a_n| <\varepsilon. $$

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  • $\begingroup$ your just saying that if the two series converges.. but our professor needs a deeper explanation. I think he's referring to "when does a series converges?". @MeeSeongIm $\endgroup$ – Crunchy Aug 27 '17 at 13:00
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    $\begingroup$ @JefferDaveCagubcob See the addendum above. $\endgroup$ – Mee Seong Im Aug 27 '17 at 13:10
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    $\begingroup$ Thanks, @MeeSeongIm . Gonna internalize this first. Actually, we're working on some problem and this came up in our solution, and we're trying to figure out why it converges. $\endgroup$ – Crunchy Aug 27 '17 at 13:22

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