The wikipedia entry for Borel summation narrates the following recollection by Mark Kac, about an encounter between Emile Borel and Mittag-Leffler. This is one of my favourites.
"Borel, then an unknown young man, discovered that his summation method gave the 'right' answer for many classical divergent series. He decided to make a pilgrimage to Stockholm to see Mittag-Leffler, who was the recognized lord of complex analysis. Mittag-Leffler listened politely to what Borel had to say and then, placing his hand upon the complete works by Weierstrass, his teacher, he said in Latin, 'The Master forbids it'."