L

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LHôpitals Rule
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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,

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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,

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Notation

• Implicit Form
• a1, a2, a3,
• Explicit Forms

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Explicit to Implicit

1. Convert the sequence to implicit form.
2. Given the function , write the
   implicit form of the sequence .

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Implicit to Explicit

1. Write the sequence in
   explicit form.
2. Write the sequence in
   explicit form.

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The Fibonacci Sequence

• Defined by the rules
• F1 1
• F2 1
• Fn2 Fn Fn1
• Implicit Form
• 1, 1, 2, 3, 5, 8, 13, 21, 34, 55,
• Fibonacci Numbers in Nature

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The Big Question

• Once again, its 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.

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Rigorous Definition
If, for ? gt 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).
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Convergence Theorem
Let f be a function defined for x ? 1. If
and ak f (k) for all k ? 1,
then
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Algebra with Limits
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The Squeeze Theorem
Suppose that ak ? bk ? ck for all k ? 1 and
that Then