How long is eternity?
Sleeping Beauty was cursed by an evil witch to be in her eternal slumber. When the Prince arrived to give her the true love kiss, however, she didn't wake up as in the original version of the story. Instead, the laughing witch magically appeared with pleasure to spoil the romance before revealing that the Prince's first kiss only brought half of his lover's soul back to the body while the other half remained in the underworld.
The witch went on explaining that he could continue giving her the true love kiss once in every other day. On day 2, his second love kiss would bring \(1/3\) of the half soul in the underworld back to beauty's body. On day 3, his third kiss would bring \(1/4\) of the remaining soul in the underworld back to the body and so on.
Finally, the witch did the math for him and claimed that it would take him "eternity" to fully bring his lover's soul back in one piece. Nonetheless, the Prince still stayed with the Sleeping Beauty to give her one true love kiss everyday despite knowing that his actions were futile.
Until one day, after giving his usual love kiss, the Sleeping Beauty's soul was 99.99% restored in her body, and seeing through his everlasting effort, the witch finally got softened and willingly lifted the spell, returning the rest of the soul back to the body. Soon afterwards, the Princess woke up, and they all lived happily ever after!
On what day did the Princess wake up? (For example, if you think it's day 10, give 10 as an answer.)