9.1: Sequences

9.1: Sequences

1. 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

2. Chapter 9: Infinite Series

3. 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

4. 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

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