Infinite Sequences

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Monotonic Convergence

Some definitions:

A sequence is bounded if $|a_n|$ never grows beyond a fixed size $M$. In other words, there is a bound $M$ such that every term in the sequence has size less than $M$.

A sequence {$a_n$} is strictly increasing if each term is bigger than the previous term. That is, $a_{n+1} > a_n$. It is non-decreasing if $a_{n+1} \ge a_n$.

Strictly decreasing means $a_{n+1}< a_n$ for all $n$, and non-increasing means $a_{n+1} \le a_n$.

If a sequence is either non-increasing or non-decreasing, it is called monotonic.

A word of caution: The terms increasing and decreasing are dangerously ambiguous, since some authors use them to mean "strictly increasing" and "strictly decreasing", while others use them to mean "non-decreasing" and "non-increasing".

Two important Theorems:

Monotonic Convergence Theorem: If a sequence is monotonic and bounded, if converges.

Unboundedness Theorem: If a sequence is not bounded, it diverges.

Notice: If a sequence is bounded but not monotonic, it might converge or it might diverge. For example,

1, -1, 1, -1, 1, -1, ... diverges

$\displaystyle 1,\, -\frac12,\, \frac13,\, -\frac14,\, \frac15,\, \ldots$ converges.

The Fundamental Theorem of Calculus

The Indefinite Integral and the Net Change

Substitution

Area Between Curves

Volumes

Integration by Parts

Integrals of Trig Functions

Trig Substitutions

Partial Fractions

Strategies of Integration

Improper Integrals

Differential Equations

Models of Growth

Infinite Sequences

Infinite Series

Integral Test

Comparison Tests

Convergence of Series with Negative Terms

The Ratio and Root Tests

Strategies for testing Series

Power Series

Representing Functions as Power Series

Taylor and Maclaurin Series

Applications of Taylor Polynomials

Partial Derivatives

Multiple Integrals

Iterated Integrals over Rectangles

Double Integrals over General Regions

Monotonic Convergence

Some definitions:

Two important Theorems: