The real line as a special union of open balls Let $\{x_k\}\subset\mathbb{R}$ be countable and dense. Let $(r_k)$ be a sequence of positive reals with $r_k\longrightarrow 0$.
Let $B_{r_k}(x_k)$ denote the open ball with radius $r_k$ around $x_k$.
Must it be that $\bigcup\limits_{k}B_{r_k}(x_k)=\mathbb{R}$?
Clearly in this case the series $\sum\limits_{k}r_k$ does not converge as otherwise it would provide a finite bound on the measure of the whole of $\mathbb{R}$. What if $\sum\limits_{k}r_k=\infty$?
 A: No, it is not necessarylly $\mathbb{R}$. 
For example, let $(x_k)_{k \in \mathbb{N}}$ be enumeration of $\mathbb{Q}$, and let $r_k = \frac{1}{2^k}$. Then $\sum_{k=1}^{\infty} r_k = 1$. Thus, the measure $\mu (\bigcup B_{r_k}(x_k))\leq 1$, and thus cannot be the whole of $\mathbb{R}$.
Even if the sum is infinite, we can not hope for the statement to be true, as the countable set could have accumulation points, such that the "big" parts of the sum just go to the accumulation points. 
This is for the unedited question.
The convergence can be as slowly as you like.
Given any series $\sum r_k$ of positive real numbers, we construct a countable set $X$, such that $\bigcup_{x \in X} B_{r_k}(x) = \mathbb{R}$.
We take $0$ as the first point. Thus we have the ball $B_{r_1}(0)$. Now, let $n_1$ be the first number such that $\sum_{k=1}^{n_1}r_k =2$. For each of those $r_k$, we add a ball on the right of our already covered subset of $\mathbb{R}$, and put a ball there. Thus we cover at least the interval $[ 0,1]$.
Now take the least $n_2$ such that $\sum^{n_2}_{k=n_1} = 2$, and repeat the same on the left side of the already covered subset. 
We can repeat this forever, choosing $(n_2, n_3, \ldots)$, as the series $\sum r_k$ is divergent. 
In each step, we cover an interval of length at least 1. 
Thus we can find a countable subset $X$ of $\mathbb{R}$, such that $\bigcup _{x \in X} B_{r_k}(x) = \mathbb{R}$.
A: The answer to the edited question is no. Consider the harmonic series $\sum \frac{1}{n}$ and rationals $\mathbb{Q}$.
Write $\mathbb{N}=\{a_1,a_2,\cdots\}\sqcup\{b_1,b_2,\cdots\}$ so that $(a_i)$ and $(b_j)$ are both increasing sequences and for which the sum of reciprocals $\sum_{i=1}^\infty 1/a_i$ converges. Then write $\mathbb{Q}=\{u_1,u_2,\cdots\}\sqcup\{v_1,v_2,\cdots\}$ satisfying the properties $|u_i|>1$ and $|v_j|\le1$. Then $\frac{1}{a_1}+\frac{1}{b_1}+\frac{1}{a_2}+\frac{1}{b_2}+\cdots$ diverges and $\{u_1,v_1,u_2,v_2,\cdots\}$ is an enumeration of the rationals but $\big( \bigcup_{i=1}^\infty B(\frac{1}{a_i},u_i) \big) \cup \big(\bigcup_{j=1}^\infty B(\frac{1}{b_j},v_j)\big)$ is proper in $\mathbb{R}$.
