In practice, absolute summability of hedging errors may not be applicable. Mostly, for the sequences of hedging errors, one relaxes the absolute convergence criteria and uses the squared summability of hedging errors. Note: Absolute summability is a stricter condition than squared summability. Some sequences may not be absolute summable but are only squared summable.
Also, to notice the significant impact of the error and to make it visible is one of the prime reasons for using sum of squares for hedging errors. Remember, we're talking about managing or controlling risks, the finer we notice the errors, the useful it shall be. :)