The advantage of the current system is that you can guess the weight in your hand, and usually can intuit how much postage to put on your envelope. (But not at the thresholds — which is what happened to me this lazy Saturday morning — forcing me to put double postage on an envelope “just to be safe.”)

Two inconveniences arise from pricing postage as a function of weight: we’re penalized for a tiny increase in weight, and we’re generally inconvenienced for having to weigh our mail before putting on postage. Okay, you avoid the former by making postage a continuous function of weight, but that increases the latter inconvenience.

But we can entirely avoid the latter by adding this optional element: Anyone can open an account with USPS and order personalized stamps with bar codes on them. (As before, we buy stamps from the post office with our credit card … but the credit card accesses our account). We slap exactly one onto each letter, regardless of the weight of the letter, and let USPS weigh our letters and charge our accounts. (They have to weigh every letter under any system, anyways.)

Make the bar codes invisible except under ultra-violet, to reduce fraud. And we get emailed a statement at the end of the month with a record of our account activity.

Problems: it really has to be optional, for obvious reasons. Maybe we can have all these systems concurrently, including the current system, and let people choose based on convenience.

Also! I’d like to point out that in your scenario, it wouldn’t matter if the postage rates keep rising: r represents the fraction of the weight of the package that must be postage. So the solution is simple: just make the “postage tape” more expensive, and r can remain constant and the sum remains small. They could change the color or something when the rate was changed.

PS: I think that’s the best spam screener I’ve ever seen.

People have trouble grasping dimensionless quantities though. You can be sure that if your scheme were implemented there’d be a name for the unit, although the name would just be an alias for “1″!

Also, the whole infinite series thing can be short-circuited by specifying the ratio of stamp weight to original package weight. You’d first weigh it, multiply by the postage rate, and cut out that quantity of tape.

But how can the post office tell that the right weight has been affixed, except by measuring the area?