A stream of perpetual future payments does not need to be decrease exponentially (or decrease at all) to have finite present value. A contract to receive $1/year forever (which you can resell or leave as inheritance etc.) is worth 1/r, where r is the risk-free rate of return at the time the contract is sold. At 1% interest a $1 perpetuity is worth $100 — the amount of money you’d have to put in a bank today to expect to get $1 in interest per year forever.

Yes, the exponential decrease is sufficient, but not necessary.

Of course, you are putting an exponential decrease into the present value calculation.

To give a truly non-exponential example: summing up 1/n^2 also only gives you a finite value.