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Mindmap von Meri perkins, aktualisiert more than 1 year ago
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Erstellt von Meri perkins
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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
- Two bounding sequences only have to "squeeze in" for small n value they may not bound the third sequence
- if an<cn<bn "For all n large enough" then cn will also have this limit
- Squeezing theorem
- Checking for convergence
- No finite value: Diverges
- If a finite value it converges
- approaches finite value
- Limit of the sequence
- 1. What does N-th term look like?
- 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
- find a formula for an to find the outcome of a n-th term
- Will have a few base terms to define outcome of other terms
- Two Important Ideas to consider
- Idea: A sequence is a list of numbers
- 1. Find Formula for n-th Term
- Chapter 9: Infinite Series
- Checklist of Key Ideas:
- Infinite sequence
- Infinite number of terms
- Infinite number of terms
- Terms of sequence
- a(n)
- A Pattern of numbers to infinity
- A Pattern of numbers to infinity
- a(n)
- Graph and limit of sequence
- Converges, or diverges
- 1/n-> limit to 0
- 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
- {1+(-1/2)^n} will occillate to 1, still converging
- {n/n+1} has a limiting value of 1
- ({(-1)^n+1} will osscillate
- Converges, or diverges
- Recursion Formulas
- Infinite sequence
- 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
- If terms are remaining constant, or becoming more positive, increasing
- Strictly Monotone
- Strictly Increasing and decreasing is when no two terms are remaining constant
- If terms gave a bound, then they converge
- If terms gave a bound, then they converge
- Strictly Increasing and decreasing is when no two terms are remaining constant
- Increasing or Decresing
- 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
- To shift indices: Replace k with j+3
- 1/k is tricky this is a harmonic series
- Actual Values can be found with geometric sequences and Telescoping sums
- a/1-r
- must start at k=0
- geometric series from k=0 to infinity ar^k
- n to infinity to see if converges
- Sn is partial sum
- Sn=sigma from 1 to n uk
- Convergence Tests
- Integral Tesr
- Use b instead of infinity and plug infity back into integral later
- Use b instead of infinity and plug infity back into integral later
- p series
- if p is greater than 1, then will diverge
- if p is greater than 1, then will diverge
- Integral Tesr
- Sum of infinitely many terms, aka, a sequence is an infinite series series
- Monotone:
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