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L’H ô pital’s Rule

Discover what sequences are, how they are defined, converted between implicit and explicit forms, and determine convergence or divergence. Learn about the famous Fibonacci sequence and its presence in nature. Explore the connection between sequences, limits, and real numbers.

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L’H ô pital’s Rule

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  1. L’Hôpital’s Rule

  2. What is a sequence? • An infinite, ordered list of numbers. {1, 4, 9, 16, 25, …} {1, 1/2, 1/3, 1/4, 1/5, …} {1, 0, 1, 0, 1, 0, –1, 0, …}

  3. What is a sequence? • A real-valued function defined for positive (or non-negative) integer inputs. {an}, where an= n2 for n = 1, 2, 3, … {ak}, where ak= 1/k for k = 1, 2, 3, … {aj}, where aj= cos((j-1)/2) for j = 1, 2, 3, …

  4. Notation • Implicit Form {a1, a2, a3, …} • Explicit Forms

  5. Explicit to Implicit • Convert the sequence to implicit form. • Given the function , write the implicit form of the sequence .

  6. Implicit to Explicit • Write the sequence in explicit form. • Write the sequence in explicit form.

  7. The Fibonacci Sequence • Defined by the rules: F1 = 1 F2 = 1 Fn+2 = Fn + Fn+1 • Implicit Form: {1, 1, 2, 3, 5, 8, 13, 21, 34, 55, …} • Fibonacci Numbers in Nature

  8. The Big Question • Once again, it’s this: convergence or divergence? • Let {ak} be a sequence and L a real number. If we can make ak as close to L as we like by making k sufficiently large, the sequence is said to converge to L. • Otherwise, the sequence diverges.

  9. Rigorous Definition If, for  > 0, there is an integer N such that then the sequence {ak} is said to converge to the real number L (i.e., {ak} has the limit L).

  10. Convergence Theorem Let f be a function defined for x 1. If and ak = f (k) for all k  1, then

  11. Algebra with Limits

  12. The Squeeze Theorem Suppose that ak bk  ck for all k  1 and that Then

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