Determine whether the sequences converge.

Notice that



So as you have . Clearly must converge.

The second sequence requires a bit more work.



The monotone convergence theorem will help here; if we can show that the sequence is monotonic and bounded, then will converge.

Monotonicity is often easier to establish IMO. You can do so by induction. When , you have



Assume , i.e. that . Then for , you have



which suggests that for all , you have , so the sequence is increasing monotonically.

Next, based on the fact that both and , a reasonable guess for an upper bound may be 2. Let's convince ourselves that this is the case first by example, then by proof.

We have




and so on. We're getting an inkling that the explicit closed form for the sequence may be , but that's not what's asked for here. At any rate, it appears reasonable that the exponent will steadily approach 1. Let's prove this.

Clearly, . Let's assume this is the case for , i.e. that . Now for , we have



and so by induction, it follows that for all .

Therefore the second sequence must also converge (to 2).

The answer would be 27 because assuming that each package has 1 bar of soap and 1 tube of toothpaste, after 27 packages they would run out of toothpaste