# Today I Learned

Some of the things I've learned every day since Oct 10, 2016

## 44: Sophmore’s Dream

Sophmore’s dream refers to the identity

$\int _0 ^1 x^{-x} \textrm{d}x = \sum _{n = 1} ^{\infty} n^{-n}$.

The name refers to the fact that this is an identity which, although true, seems ‘too good to be true’ in some sense, and looks at first like the naive misapplication of ideas not well-understood by the user. A good second example of this is the freshman’s dream, which refers to the mistaken identity $(a + b)^n = a^n + b^n$.