No, it still isn't infinite. Under reasonable assumptions the series is actually bounded from above by a geometric series [1] which is famous for having a finite sum when ratio is <1 [2]. In this case, the ratio r=1/(1+i) is indeed <1. More importantly, no asset exists an infinite amount of time, so the sum is actually finite anyway.

Perhaps we should dispose with all the math though as it should be intuitively clear that if we discount future cashflows at zero rate, i.e. if we do not discount future cashflows, then their present value should be equal to the future value. After all, discounting just means that you value future money less than today's money. If you don't discount, you value both the same, so whatever formula for discounting you use it must be true that plugging in zero rate yields NPV equal to the sum of future cashflows.

Agreed about risk discount. Perhaps what you mean by "whippy" is that as the risk-free interest rate goes to zero NPV becomes increasingly sensitive to the risk discount.

[1]: https://en.wikipedia.org/wiki/Geometric_series

[2]: See [1]. Also, as an example, recall that 1+1/2+1/4+1/8+...=2.