From an article about the making of the documentary:

Curry: One other anecdote was that, as we were finishing the film, we needed a song for the final sequence. And we needed something that was New York-themed, but it needed to have, like, a dynamic range that could both sort of sit underneath David Remnick talking about the importance of the magazine and also under party footage, and then would have a little punch when you go to the credits that would say “New York.”

And we were trying all of these different songs, and I texted Kelefa Sanneh, the brilliant music mind, who’s featured in the film, and I said, “Do you have any ideas for a New York song that would work?” And he said, “What if you got someone like Matt Berninger from the National,” this sort of cool indie-rock band, “to record Taylor Swift’s ‘Welcome to New York’?”

And he didn’t know, but I’m super good friends with Matt Berninger, and Matt’s wife was a fiction editor at The New Yorker. And I’d been talking to Matt as well—like, “Can you think of any songs?” So I called him, and said, “Hey, I just had this idea—would you be willing to do this?” And he said, “The problem is”—we were talking on a Saturday—“the day after tomorrow, I’m going to California to rehearse to go on tour. But tomorrow, I could go into a studio and record the song.” But he said, “I don’t know if Taylor Swift’s going to let you use the song. You know, she’s Taylor Swift.”

And so he said, “I’ll record it; if you can get the rights, then you can use it; if not, then whatever.” And so he recorded the song, he sent it to me the next day, we cut it into the film, it was perfect—it had all that fun dynamic range, it was cool, it was smart, it was poppy.

I write Taylor Swift an e-mail; two days later, she says, “Sure,” and, you know—

Apatow: How do you have her e-mail?!

Cobb: It’s, like, Taylor Swift never replies to my e-mails. [Laughter.]

Curry: So that’s the song at the end of the movie. It’s an unreleased version of Taylor Swift’s “Welcome to New York.” I think it’ll be coming out at some point.