alternating discrete distribution

Question

Let us take probability mass function $p = \{p_{1}, p_{2}, \dots \}$ with an infinite support. From what I can see, all discrete infinitely supported probabilities in wikipedia have the following property: starting from some number $k$ we have: $$ p_{i+1} \leq p_{i} $$

The question: is it possible to have discrete infinitely supported alternating distribution, i.e. there is no finite $k$ such that the equality above holds?

I tried to solve the problem using the Ratio test for convergent series.

Answer 1

Essentially, what you're asking is whether it is possible to have a non-monotonic sequence of positive numbers that add up to $1$. We can take the sequence $\left\{2^{-k}\right\}^{\infty}_{k=1}$ and reverse the order of each pair of terms, i.e.,

$$\left\{\frac{1}{4},\frac{1}{2},\frac{1}{16},\frac{1}{8},...\right\}.$$

This doesn't change the sum, but it does prevent the sequence from becoming monotonic after a finite number of terms.

Answer 2

Yes, an easy example would be to distribute the mass only over the even numbers.

That is, let $p_{2n} = 2^{-n}$ (for $n \geq 1$) and $p_{2n+1} = 0$.