Infinite Sequences

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• Misc
  • My solutions to exam 1 are online through “Exercises” page
  • Earbuds left in my office before break?
  • Colloquium
    • Monday March 30, 4:00-5:00 pm
    • Newton 201
    • Julie Croteau ’97
    • Department of Mathematics
    • Corning Community College
    • “Bodybuilding! ... for Mathematicians”
    • Extra credit for summaries
• Questions?
• Sequences
  • Section 10.1
    • Sequence = list of numbers
      • Can be infinite, e.g., 2, 4, 6, 8, ..., 2n, ...
    • Converge vs diverge?
      • “converge” = terms in list approach a finite limit
        • e.g., 1/n
        • 1, 1/2, 1/3, ...
        • a_n = 1/n, n ≥ 1
        • Converges: lim_n→∞a_n = lim_n→∞(1/n) = 0
      • “Diverge” if goes to infinite number
        • ...or if terms oscillate e.g., 1, -1, 1, -1, ...
    • Calculating limits
      • Sum, difference, constant multiple, product, etc. rules
      • Sandwich theorem
      • L’Hopital’s rule
      • Using continuous functions
        • If f continuous at L and L = lim_n→∞a_n then lim_n→∞f(a_n) = f(L)
      • i.e., use what you already know about limits
  • Where is this going?
    • (Infinite) “series” = sum of (infinite) sequence
    • Lots of functions can be described as fairly simple infinite series
      • e.g., R examples of functions that calculate certain (long but finite) sequences and add them up, to get results that look remarkably like the sine and exponential functions
        

        taylor1 = function( x ) {
            n = seq( 0, 50 )
            a = x^(2*n+1) / factorial(2*n+1) * (-1)^n
            sum( a )
        }

        • Looks just like sin(x) in a plot:
          
          

          taylor2 = function( x ) {
              n = seq( 0, 50 )
              a = x^n / factorial(n)
              sum( a )
          }

        • Looks just like e^x:
          
    • Consider the sequence underlying a series, e.g., a_n = xⁿ / n!, n ≥ 0 (and 0! = 1)
      • a₀ = 1
      • a₁ = x
      • a₂ = x² / 2
      • a₃ = x³ / 3! = x³ / 6
      • (x arbitrary constant)
      • Converges to 0
        • Per Theorem 5 from reading
    • Sequence diverging to infinity means corresponding series sums to infinity
    • Diverging by oscillating means series doesn’t have a defined limit
    • Converging, e.g., to 1
      • • ..., 1+ε, 1+ε, 1, 1, 1, ...
        • The sum of these will be infinite
    • Moral: series are useful and mathematically interesting, at least if they converge. Sequences and their convergence are a foundation for understanding series.
• Next
  • More sequences

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