1. Properties of the function \(x^x\) (Post-Doc Ergo Propter Hoc H/T Dave Richeson). Can you prove:

\[\int_{0}^{1} x^{x} = -\sum_{n=1}^{\infty} (-n)^{-n}\]

2. Steve Strogatz on Einstein's boyhood proofs (newyorker).

A strange derivative: let \[f(x)=x^2 \sin(1/x^2)\] for \(x \ne 0\), and \(f(0)=0\). This function has \(f'(0)=0\) even though \[\lim_{x \rightarrow 0} f'(x),\] does not exist.

\[\int_{0}^{1} x^{x} = -\sum_{n=1}^{\infty} (-n)^{-n}\]

2. Steve Strogatz on Einstein's boyhood proofs (newyorker).

The style of his Pythagorean proof, elegant and seemingly effortless, also portends something of the later scientist. Einstein draws a single line in Step 1, after which the Pythagorean theorem falls out like a ripe avocado. The same spirit of minimalism characterizes all of Einstein’s adult work. Incredibly, in the part of his special-relativity paper where he revolutionized our notions of space and time, he used no math beyond high-school algebra and geometry.3. Via Twitter,

**Steven Strogatz**@stevenstrogatzA strange derivative: let \[f(x)=x^2 \sin(1/x^2)\] for \(x \ne 0\), and \(f(0)=0\). This function has \(f'(0)=0\) even though \[\lim_{x \rightarrow 0} f'(x),\] does not exist.