Q:

What's your favorite equation, formula, identity or inequality?

Certain formulas I really enjoy looking at like the Euler-Maclaurin formula or the Leibniz integral rule.  What's your favorite equation, formula, identity or inequality?

A:

Stokes' Theorem 

A:

Trivial as this is, it has amazed me for decades:
$(1+2+3+...+n)^2=(1^3+2^3+3^3+...+n^3)$

A:

$$ \frac{24}{7\sqrt{7}} \int_{\pi/3}^{\pi/2} \log \left| \frac{\tan                
      t+\sqrt{7}}{\tan t-\sqrt{7}}\right| dt\\ = \sum_{n\geq                         
     1} \left(\frac n7\right)\frac{1}{n^2}, $$
where $\left(\frac n7\right)$ denotes the Legendre symbol. Not really 
my favorite identity, but it has the interesting feature that it is a
conjecture! It is a rare example of a conjectured explicit identity
between real numbers that can be checked to arbitrary accuracy.
This identity has been verified to over 20,000 decimal places.
See J. M. Borwein and D. H. Bailey, Mathematics by Experiment:
 Plausible Reasoning in the 21st Century, A K Peters, Natick, MA,
2004 (pages 90-91).

