# Conditions of the Sum of Infinite Summations

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?

• As long as two of the sums converge. – Angina Seng Aug 27 '17 at 12:52
• @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. – 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 $$|a_m+a_{m+1}+\ldots + a_n| <\varepsilon.$$