What was found
ON TAXI INTO THE GATE THE APU WOULD NOT START AND THE APU FAULT LIGHT ILLUMINATED. GROUNDED - YES. ACCOMPLISHED B737-900 FIM 49-40 TASK 803. NO DEBRIS IN MAGNETIC DRAIN PLUG, APU FREELY ROTATES. ACCOMPLISHED GVI OF APU HARNESS AND STARTER GENERATOR HARNESS, NO DEFECTS. STARTED AND RAN APU ON BATTERIES ONLY, START NORMAL AND HOLDS LOAD. 3 BREAKERS TAGGED/ RESET. REMOVED AND REPLACED STARTER CONVERTER UNIT AS PRECAUTIONARY, IAW B737-900 AMM 49-41-61/401. GROUND VERIFICATION TEST GOOD TWICE.