Algebraic

  • You can split summations (even infinite ones)
  • Equivalently, Pushing log thru product makes it a sum
  • Telescoping sums and products
  • Break up summations
  • Summation of Squared

Probability

  • Tower property
  • Linearity of expectation. You can switch expectation and summation
  • Definition of expectation is an integral (or summation)

Calculus

  • switch integrals

Inequalities

  • Sum greater than max
  • Max greater than averages
  • Average greater than min. Combined:

always.