How can an LED series continue to work after an LED goes bad?
Most strings of LEDs that I have seen are not a simple series.

Instead you have multiple series of 2* LEDs plus resistors in parallel:



If one LED goes open circuit then the other LED in the same line will go out.  If one LED goes short-circuit then the other LED will light up brighter, and will die sooner - if not almost immediately depending on voltages and tolerances etc.

Example:



The exact order of resistor and diodes is irrelevant - in that one the resistor comes between the diodes instead of before them, but that makes no difference at all to it electrically.

*or more depending on voltages in use