9.1: Sequences - created from Mind Map

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Chapter 9: Infinite Series

Checklist of Key Ideas:

Purpose: How to determine if they converge

9.2: Monotone Sequences

1. Find Formula for n-th Term

Idea: A sequence is a list of numbers

Two Important Ideas to consider

Recursive sequence

1. What does N-th term look like?

2. Does a sequence approach a limit and converge

Limit of the sequence

approaches finite value

If a finite value it converges

No finite value: Diverges

Checking for convergence

Squeezing theorem

if an<cn<bn "For all n large enough" then cn will also have this limit

Two bounding sequences only have to "squeeze in" for small n value they may not bound the third sequence

Will have a few base terms to define outcome of other terms

find a formula for an to find the outcome of a n-th term

Infinite sequence

Terms of sequence

Graph and limit of sequence

Recursion Formulas

Infinite number of terms

a(n)

A Pattern of numbers to infinity

Converges, or diverges

{n+1} will increase without bound

1/n-> limit to 0

({(-1)^n+1} will osscillate

{n/n+1} has a limiting value of 1

{1+(-1/2)^n} will occillate to 1, still converging

Monotone:

9.3 Infinite Series

Increasing or Decresing

Strictly Monotone

If terms are remaining constant, or becoming more positive, increasing

Decreasing when constant or more negative terms

Strictly Increasing and decreasing is when no two terms are remaining constant

If terms gave a bound, then they converge

Sum of infinitely many terms, aka, a sequence is an infinite series series

Convergence Tests

Sn=sigma from 1 to n uk

Sn is partial sum

n to infinity to see if converges

geometric series from k=0 to infinity ar^k

must start at k=0

a/1-r

Actual Values can be found with geometric sequences and Telescoping sums

1/k is tricky this is a harmonic series

To shift indices: Replace k with j+3

Integral Test

Use b instead of infinity and plug infity back into integral later

p series

if p is greater than 1, then will diverge