9.1: Sequences

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Infinite Series
Meri perkins
Mind Map by Meri perkins, updated more than 1 year ago
Meri perkins
Created by Meri perkins over 6 years ago
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Resource summary

9.1: Sequences
  1. Purpose: How to determine if they converge
    1. 1. Find Formula for n-th Term
      1. Idea: A sequence is a list of numbers
        1. Two Important Ideas to consider
          1. 1. What does N-th term look like?
            1. 2. Does a sequence approach a limit and converge
              1. Limit of the sequence
                1. approaches finite value
                  1. If a finite value it converges
                    1. Checking for convergence
                      1. Squeezing theorem
                        1. if an<cn<bn "For all n large enough" then cn will also have this limit
                          1. Two bounding sequences only have to "squeeze in" for small n value they may not bound the third sequence
                    2. No finite value: Diverges
              2. Recursive sequence
                1. Will have a few base terms to define outcome of other terms
                  1. find a formula for an to find the outcome of a n-th term
          2. Chapter 9: Infinite Series
            1. Checklist of Key Ideas:
              1. Infinite sequence
                1. Infinite number of terms
                2. Terms of sequence
                  1. a(n)
                    1. A Pattern of numbers to infinity
                  2. Graph and limit of sequence
                    1. Converges, or diverges
                      1. 1/n-> limit to 0
                      2. {n+1} will increase without bound
                        1. ({(-1)^n+1} will osscillate
                          1. {n/n+1} has a limiting value of 1
                            1. {1+(-1/2)^n} will occillate to 1, still converging
                      3. Recursion Formulas
                      4. 9.2: Monotone Sequences
                        1. Monotone:
                          1. Increasing or Decresing
                            1. If terms are remaining constant, or becoming more positive, increasing
                              1. Decreasing when constant or more negative terms
                              2. Strictly Monotone
                                1. Strictly Increasing and decreasing is when no two terms are remaining constant
                                  1. If terms gave a bound, then they converge
                              3. 9.3 Infinite Series
                                1. Sum of infinitely many terms, aka, a sequence is an infinite series series
                                  1. Sn=sigma from 1 to n uk
                                    1. Sn is partial sum
                                      1. n to infinity to see if converges
                                        1. geometric series from k=0 to infinity ar^k
                                          1. must start at k=0
                                            1. a/1-r
                                              1. Actual Values can be found with geometric sequences and Telescoping sums
                                                1. 1/k is tricky this is a harmonic series
                                                  1. To shift indices: Replace k with j+3
                                  2. Convergence Tests
                                    1. Integral Tesr
                                      1. Use b instead of infinity and plug infity back into integral later
                                      2. p series
                                        1. if p is greater than 1, then will diverge
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