The elders make the argument that the centurion is a friend to the Jews who donated money to the synagogue, which convinces Jesus to go and heal his servant. While he's on the way, the centurion's friends stop them because he hasn't had time to vacuum. He asks Jesus to heal the servant from outside. Somehow, Jesus interprets this to mean that the centurion is the most faithful man in Jerusalem. When the friends get back, they discover the slave has recovered.
The next day, Jesus goes to Nain, where he finds a funeral in progress. The young man is his widowed mother's only child. Jesus goes up to the casket, touches it and says Young man, I say unto thee, Arise (v. 14). The guy does, and says BRAINS!. Is this Lazarus?
Anyway, raising a zombie makes the people believe that Jesus is a new prophet. Rumours quickly spread all over Judea, even reaching John the Baptist in prison. John sends two of his followers to ask if Jesus is the real deal, or if they should keep waiting. So Jesus cures a bunch of people in front of them, then tells them to go back to John and report their findings.
Jesus turns to his own followers and says that John is the person the scriptures predicted, and he's a good prophet, for a human, but he's nothing compared to what you find in heaven. The people who have been baptised lap this up, but the unbaptized Pharisees and lawyers are skeptical.
Then he starts criticising the current generation, like all old farts everywhere, in this case for failing to recognise John the Baptist's or his own awesomeness.
The Pharisees decide that if they can't beat him, they should join him, so they invite Jesus over for a meal. While he's sitting there, a woman comes along with a box of ointment. Then she stands behind Jesus, whining and snivelling and washing his feet with the tears and drying them with her hair and rubbing them with ointment.
The Pharisee is watching this, and like any sane person, thinks it's weird. Jesus listens in on his thoughts and tells him a weird parable about a man who has two debtors, one who owes him 500 pence, and another who owes him 50. He forgives them both. Which one is more grateful? Given that gratitude is the shortest-lived human emotion neither of them. Nevertheless, the Pharisee, now called Simon, says the man who owes 500 pence will love his creditor more. Jesus says that's correct, then explains the moral: this woman is a very sinful person, being a woman, but because she has shown him much love, he will grant her much forgiveness. But Simon, who has not been very loving, except for the free food, will not be forgiven for most of his sins. This gives the Pharisees new reasons for their skepticism, but at least the woman leaves happy.