This is a note-to-self page which I’ll update as I naturally revisit these ideas opportunistically.

Special numbers

: alternating signs through odd/even powers
: null (trivial additive solution), invariant (sums to zero)
: identity (trivial multiplicative solution), invariant (multiplies to 1)
shrinks with growing powers
:
Odd: , Even:

Problem solving approaches

Properties of linearity, aka superposition

Find ways to see a raw definition of a concept hidden in the problem you’re solving.

Plugging in easy/obvious examples to verify a hypothesis (often used in differential equations) during exploration

Famous functions

Things only a constant function can do

Small goes to is the same as the small angle approximation for

Quadratics: Exploit (e.g. used in trace and det to infer eigenvalues)

Calculus

Symmetric integrals cancels for odd function and doubles of one side for even functions

Series

Spotting hidden famous series (such as geometric sums)

Series expansion dropping terms

is even terms of with alternating signs starting with 1,
is odd terms of with alternating signs starting with

Taylor series always have factorial at the bottom (denominator) of the coefficient matching the n-th derivative at the top (numerator) for the n-th power term.

Telescoping series (adjacent terms cancels)

Use derivative to bring down polynomial power by and create a shifted series (which can be used to recurse or cancel)

Topology

In real line topology, outside the intuitive examples (singletons included), consider universal and empty set first, rationals and irrationals, then blame Cantor.

Discrete Math (or Primes)

Modulos: generate all possible remainders of a certain modulo by multiplying.