Mathematical Tools II: Taylor Series Expansion

These flashcards cover key terms and concepts regarding Taylor series expansion and related mathematical definitions.

Negligible

The function f is called negligible compared to g around a if f(x) = o(g(x)) as x approaches a.

Equivalent Functions

The functions f and g are equivalent around a if f(x) ∽a g(x) as x approaches a.

Reflexive Relation

A relation f ∽a f, meaning a function is equivalent to itself.

Symmetric Relation

If f ∽a g, then it implies g ∽a f, showing mutual equivalence.

Transitive Relation

If f ∽a g and g ∽a h, then it implies f ∽a h, showing a chain of equivalence.

Taylor Series Expansion

A representation of a function as an infinite sum of terms calculated from the values of its derivatives at a single point.

Polynomial Part

In a Taylor series expansion, the polynomial part is denoted as Pn(x - a).

Error Term

In a Taylor series expansion, the term (x - a)ⁿε(x) is known as the remainder or error term.

Asymptotic Taylor Series

A Taylor series expansion at +∞, meaning the function behaves like a polynomial as x approaches infinity.

Growth Comparisons

ln(x) = +∞ o(x^r) indicates logarithmic growth is negligible compared to polynomial growth.

Function of Class C n

A function that is n times continuously differentiable.

Unique Expansion

The Taylor series expansion of function f around point a of order n is unique.

Even Function

A function with a Taylor series expansion around 0 containing only even degree terms.

Odd Function

A function with a Taylor series expansion around 0 containing only odd degree terms.

Anti-derivative

A function F that results in f when differentiated.