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13038511
9.1: Sequences
Descripción
Infinite Series
Sin etiquetas
calc 2
Mapa Mental por
Meri perkins
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Resumen del Recurso
9.1: Sequences
Purpose: How to determine if they converge
1. Find Formula for n-th Term
Idea: A sequence is a list of numbers
Two Important Ideas to consider
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
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
No finite value: Diverges
Recursive 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
Chapter 9: Infinite Series
Checklist of Key Ideas:
Infinite sequence
Infinite number of terms
Terms of sequence
a(n)
A Pattern of numbers to infinity
Graph and limit of sequence
Converges, or diverges
1/n-> limit to 0
{n+1} will increase without bound
({(-1)^n+1} will osscillate
{n/n+1} has a limiting value of 1
{1+(-1/2)^n} will occillate to 1, still converging
Recursion Formulas
9.2: Monotone Sequences
Monotone:
Increasing or Decresing
If terms are remaining constant, or becoming more positive, increasing
Decreasing when constant or more negative terms
Strictly Monotone
Strictly Increasing and decreasing is when no two terms are remaining constant
If terms gave a bound, then they converge
9.3 Infinite Series
Sum of infinitely many terms, aka, a sequence is an infinite series series
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
Convergence Tests
Integral Tesr
Use b instead of infinity and plug infity back into integral later
p series
if p is greater than 1, then will diverge
Recursos multimedia adjuntos
Convergence Tests (binary/octet-stream)
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