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AP Calculus II/BC > Sequences
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QuickNotes™  

Sequences

Main definitions and theorem:

Let {an} be a sequence.

Definitions:

  • {an} is monotonically increasing means an+1an for all n.

  • {an} is monotonically decreasing means an+1an for all n.

  • {an} is monotonic means {an} is one or the other.

  • {an} is bounded means there exists M such that |an| ≤ M for all n.

Theorem: If {an} is bounded and monotonic, then it converges.

Hints and tips:

  • In determining whether a sequence converges or diverges and what it converges or diverges to, you can ignore the first few terms. What is important is what the later terms do.

  • Don’t try to analyze sequences by plugging numbers into a calculator. This is extremely unreliable.

  • You often use the “bounded monotonic” theorem to analyze sequences that are defined recursively, that is, an+1 is defined in terms of an.

  • When analyzing fractions, focus on the biggest term in the top and the biggest term in the bottom. Remember the following ranking of functions:

  • You can use L’Hôpital’s Rule for situations of the form or . If you have 0 · ∞, you must put it into fraction form before you can use L’Hôpital. The same goes for 1 : Write . Then you can often sort out the exponent using L’Hôpital.

  • When you have an expression of the form a − b, especially when square roots are involved, it is often useful to multiply by to take advantage of the identity (a − b)(a + b) = a² − b² .

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