I have recently acquired a second-hand iPhone 5s. The battery needs recharging at least twice a day, with only moderate use. I find that the phone gives me no warning of impending battery failure, but when indicating that there is still about 50% charge just suddenly goes "black-screen." When put on charge it fairly quickly displays a similar amount of charge and then climbs slowly to 100%. Am I right, do you think, in assuming that replacing the battery will solve the problem of sudden failure, and short battery life, or is this an indication of some other fault in the phone? Thanks.