Jumping in two-dimensional space? I came up with this problem and have no idea how to approach it: assume that a bug starts at $(0,0)$ and at every second, it jumps in one of the four directions. At second $i$, it jumps a distance of $a_i$, where $a_1,a_2,a_3 \dots$ is a converging series with positive terms. For example, $a_i = \dfrac{1}{3^i}$ or $a_i = \dfrac{1}{i^2}$.
What is the area of all possible locations the bug can reach? Is it even a continuous area? Is there any general way to approach this problem, or is it dependent on the sequence?
I also considered the simpler problem of what happens if the bug can only move up or right; in that case, its the line $x+y=S$ in the first quadrant, where $S$ is the sum of the sequence. Can you always achieve every point on the line? If $a_i = \dfrac{1}{c^i}$, where $c$ is an integer, you can just use the base $c$ representation to construct anything. Other than that, I have no progress. 
I'd love to hear if you have any interesting ideas!
 A: This is not a full answer but it may be helpful.
If the series were divergent but with $a_n$ going to zero, then you would be able to reach every point on the plane. This can be proven with an argument in the style of the standard proof of Riemann's rearrangement theorem.
If the series is convergent to a sum $S$ then for any reachable point $(x,y)$ you must have $|x| + |y| \leq S$. This gives an upper bound of $2 S^2$ to the reachable area.
I don't know if the reachable region can have positive measure, but I'm sure that you can't give a lower bound for it using the sum of the series. Take a series $\sum_{n\geq1}{a_n}$ of positive terms converging to $S + \epsilon$ such that $a_1 = S$. Then, with the first step, you can move $S$ units in any of the four directions. After that step you have a problem like the original one but around any of the points $(0,S), (0,-S), (S,0)$ or $(-S,0)$. Around each of these points, you can reach a maximum area of $2\epsilon ^2$ so the total reachable area will be at most $8\epsilon ^2$. Because $S$ and $\epsilon$ were arbitrary, you can't bound below the reachable area with a non-trivial expresion involving $S+\epsilon$ alone.
Reasoning like before but with $N$ steps first and then bounding, you get an upper bound for the reachable area of $4^N 2(\sum_{n\geq N+1}{a_n})^2 = 2 (2^N(S-S_N))^2$ where $S_N$ is the $N$-th partial sum of the series.
This will solve the problem is the series is fast convergent. More precisely, you can say that the area of the reachable region is zero if $2^N(S-S_N) \to 0$.
Observe that if there is a chance to reach positive area, you will need to give an infinite number of horizontal and vertical steps to cover such a region. For example, if you permit only a finite number of horizontal steps, then there are countable may possibilities for the $x$ coordinates of the reached points. Thus, the reached region is contained in a countable union of vertical lines and it has zero measure because of that.
For a general series that is not so fast convergent I'm not sure what to expect. It's a very nice problem.
