$\sum_{n=1}^{\infty} (a_n+b_n)=\sum_{n=1}^{\infty} a_n + \sum _{n=1}^{\infty} b_n$? It is true to say that $\sum_{n=1}^{\infty} (a_n+b_n)=\sum_{n=1}^{\infty} a_n + \sum _{n=1}^{\infty} b_n$? It seems false but i can't find any counterexample..
 A: If $\sum a_n$ and $\sum b_n$ converge, then $\sum (a_n+b_n)$ converges to $\sum a_n+\sum b_n$.
If $\sum a_n$ converges but $\sum b_n$ diverges, or if $\sum a_n$ diverges and $\sum b_n$ converges, then $\sum (a_n+b_n)$ diverges.
If $\sum a_n$ and $\sum b_n$ diverge, then $\sum (a_n+b_n)$ may converge or diverge.
A: One helpful point might be to think in terms of the partial sums. Let $A_N = \sum\limits_{n=1}^N a_n$, $B_N = \sum\limits_{n=1}^N b_n$, and $C_N = \sum\limits_{n=1}^N (a_n + b_n)$. It is clear for every $N \in \Bbb{N}$ that $A_N + B_N = C_N$. Now, let's apply the theory of sequences.
You know that if $A_N$ converges as $N \to \infty$ and $B_N$ converges as $N \to \infty$ then $A_N + B_N$ converges and $\lim (A_N + B_N) = \lim A_N + \lim B_N$, therefore $\lim C_N = \lim A_N + \lim B_N$.
The other cases can be handled similarly. Such as if $A_N \to \infty$ and $B_N$ converges then $A_N + B_N \to \infty$.
A: A counter example is the following:
$$
a_n=\dfrac{1}{n},
$$
$$
b_n=\dfrac{-1}{n+1},
$$
So you have:
$$
\begin{equation}
\begin{split}
\sum_{n=1}^{\infty}\left(a_n+b_n\right)&=\sum_{n=1}^{\infty}\left(\dfrac{1}{n}-\dfrac{1}{n+1}\right),\\
&=\underbrace{\sum_{n=1}^{\infty}\left(\dfrac{1}{n(n+1)}\right)}_{\mathrm{converges}},\\
&\neq \sum_{n=1}^{\infty}a_n+\sum_{n=1}^{\infty}b_n,\\
&\neq \underbrace{\sum_{n=1}^{\infty}\dfrac{1}{n}}_{\mathrm{diverges}:\,\\\mathrm{Harmonic\; series}}-\underbrace{\sum_{n=1}^{\infty}\dfrac{1}{n+1}}_{\mathrm{diverges}:\,\\\mathrm{Harmonic\; series}}.
\end{split}
\end{equation}
$$
A: Actually is true if $\ \sum _{n=1}^{\infty }\:a_n$ and $\ \sum _{n=1}^{\infty }\:b_n$ converges. 
For example, let $a_{n}=1$ and $b_{n}=-1$ for all $n\in \mathbb{N}$, then $\ \sum _{n=1}^{\infty }\:a_n=\ \sum _{n=1}^{\infty }\:1$ and $\ \sum _{n=1}^{\infty }\:b_n=\ \sum _{n=1}^{\infty }\:-1$ doesn't converge, but $\ \sum _{n=1}^{\infty }\:a_n+b_n=\ \sum _{n=1}^{\infty }\:0=0$ converges.
The answer provided by Jasper should give you the reason. Just write if you need a proof.
