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Sequences

Main definitions and theorem:

Let {a_n} be a sequence.

Definitions:

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

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

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

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

Theorem: If {a_n} 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, a_n+1 is defined in terms of a_n.

• 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 .