Question: I have two null sequences $(a\_n)\_{n\in\mathbb N}$ and $(b\_n)\_{n\in\mathbb N}$ with $a\_n \ge 0$ and $b\_n \le 0$ for all $n\in \mathbb N$. Let $(c\_n)\_{n\in\mathbb N}$ be a sequence with either $c\_n=a\_n$ or $c\_n=b\_n$. I do not know which member of $c\_n$ is actually $a\_n$ and which is $b\_n$. I know, that until the n-th member $k(n)$ members of $(c\_n)\_{n\in\mathbb N}$ are from $(a\_n)\_{n\in\mathbb N}$ and $n-k(n)$ members are from $(b\_n)\_{n\in\mathbb N}$.

**My Question:** When does the series $\sum\_{l=0}^{\infty} c\_l$ converges? Is there a good convergence test for this problem?

**My Attempt:** The expectational value of $c\_n$ should be

$$E(c\_n) = \frac{k(n)}{n} a\_n + \frac{n-k(n)}{n} b\_n$$

So I think, that $\sum\_{l=0}^{\infty} c\_l$ converges iff $\sum\_{l=0}^{\infty} E(c\_l)$ converges. Is this right? Can you point me to a theorem proving this?

Answer: 
Consider the sequence of functions on $(0,1)$
$$f\_n(x) = \begin{cases}
n & \text{ if } x \in (0,1/n)\\
0 & \text{ otherwise}
\end{cases}$$
We have $\lim\_{n \to \infty} f\_n(x) = 0 = f(x)$. However,
$$\lim\_{n \to \infty} \int\_0^1f\_n(x)dx = 1 \neq 0 = \int\_0^1 f(x) dx$$
The key in the dominated convergence theorem is that sequence of functions $f\_n(x)$ must be dominated by a function $g(x)$, which is also integrable.
It is not only a matter of being dominated by just any function $g$, but by an integrable function. $|x|$ is not integrable on $\mathbb{R}$, as $\int\_{-\infty}^\infty |x| dx = \infty$. 

copper.hat provided an example that is not dominated by any function, but you can also pick bounded examples, such as $f\_n = 1\_{[n,n+1]}$.