Karteikarten: Infinite Series and Taylor Methods — 11 Karten

What is an infinite sequence in mathematics?

An infinite sequence is an ordered list of numbers indexed by integers.

Sequence limit

Sequence terms approach L as n→∞.

When does a sequence a_n converge to a limit L?

When its terms become arbitrarily close to L as n becomes large.

Monotone + bounded

Always convergent sequences.

What defines a monotonic sequence?

It is either increasing with a_n < a_(n+1) or decreasing with a_n > a_(n+1).

Geometric sum convergence

Converges if |r|<1; sum = a/(1-r).

What does every bounded monotonic sequence do?

It converges.

Series divergence test

Diverges if terms don't tend to zero.

Partial sum

Sum of first n terms of series.

Power series

Infinite sums with variable x.

Taylor series

Series expansion of functions about a point.