Essay Series: Sequences a Sequence

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November 2013
MATH 102-01
Test III Study Guide
Chapter 8: Series

8.1 Sequences A sequence is a list a1, a2, a3, a4,…, an, … Written in a definite order. We call a1 the first term, a2, the second term, an, the nth term, etc
Notation

Bounded Sequences A sequence {an} is bounded above if there is a number M such that We say that {an} is bounded below if there is a number m such that A sequence {an} that is bouned above and bounded below is called a bounded sequence.

Monotonic Sequences A sequence {an} is increasing if an < an=1 for all n ≥ 1. A sequence {an} is decreasing If an > an+1 for all n ≥ 1. A sequence is monotonic if it is either increasing or decreasing.

Verifying Monotonicity: - Define a function f(x) for all (positive) real numbers such that an = f(n) for n an integer. If f is increasing (check f’), then an is increasing. (Same for decreasing.) - Compare consecutive terms.

- If those are inconclusive, look at first few terms. This can tell you the sequence is not monotonic. However, it cannot show that it is monotonic.

Convergence A sequence {an} has the limit L and we write

if we can make the terms an as close to L as we like by taking n sufficiently large. If exists, we say the sequence converges (or is convergent). Otherwise, we say the sequence diverges (or is divergent).

Relationship to Limits of Functions

Important consequence: You can use L’Hospital’s Rule to help…