Arithmetic growth versus exponential decay I have a kilogram of an element that has a long half-life - say, 1 year - and I put it in a container. Now every day after that I add another kilogram of the element to the container.
Does the exponential decay eventually "dominate" or does the amount of the substance in the container increase without bound?
I know this should be a simple answer but it's been too long since college...

On the first day, you have 1 kilogram of decayium. On the second day, that 1 kilogram decays into $1/\sqrt[365]{2}$ kilograms of decayium, on the third day you have $1/(\sqrt[365]{2})^2$ kilograms from the original... And so on, until after a year has passed, you finally have $1/(\sqrt[365]{2})^{365} = 1/2$ kilograms form the original. Seems legit up until here.
But everyday, you're adding 1 kilogram to whatever already exists, so you have multiple copies of this sequence all running in parallel. So on the second day, there are $1 + 1/\sqrt[365]{2}$ kilograms, on the third day, $1 + 1/\sqrt[365]{2} + 1/(\sqrt[365]{2})^2$, ... and on the $n$th day, you will have
$$\sum_{i=0}^{n-1} \frac{1}{(\sqrt[365]{2})^i}$$
kilograms of decayium.
This is a geometric sequence, which we know has a sum of $\frac{1 - 1/(\sqrt[365]{2})^n}{1 - 1/\sqrt[365]{2}}$. As $n$ becomes larger and larger (tends to infinity), this actually converges: it approaches $\frac{1}{1 - 1/\sqrt[365]{2}} = 527$ (approximately). So it becomes a stable system of sorts, which means that you'll keep getting closer to 527 kilograms no matter how long you continue adding material.
