Construct an infinite series of positive terms where sumi ai is finite yet sumi aip is infinite for every exponent 0 < p < 1. I've constructed one, though it's rather ugly. I was hoping there's a simple example I'm missing.
A simple near-example is ai = 1/i. The sum diverges for 0 < p <= 1 (from the p-series test, you could say), which is just a little too much divergence. I tried a number of things to modify this series to get convergence at p=1 without messing up the rest, though I had no luck.