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Sequences { S_n } , { a_n } - - Limits and Laws

In a Nut Shell: A sequence, {a_n} is a set of ordered numbers. The graph shown

below illustrates a sequence of ordered numbers.

For this sequence a₁ = 1, a₂ = ¼, a₃ = 1/9, etc

As we take more terms of the sequence it may or may not approach a limit. If there

is no limit, then the sequence diverges. Else, the sequence converges. The limit of the

sequence shown above converges to zero.

Limit of a Sequence, { a_n }

lim a_n = L provided a_ncan be made as close to L by picking n sufficiently large n.

n→ ∞

If there is no limit, then the sequence diverges.

Basic Limit Laws are as follows:

If lim a_n= A and lim b_n= B, and C = constant, then

n→ ∞ n→ ∞

lim Ca_n= CA , lim (a_{n +}b_n)= A + B, lim a_nb_n=AB , lim a_n/b_n= A/B

n→ ∞ n→ ∞ n→ ∞ n→ ∞

Squeeze Law for Sequences – Use the squeeze law when the sequence is complicated

or changes wildly.

If { a_n } and { c_n } each converge to L and if a_n ≤ b_n ≤ c_n for

all n beyond some point, then lim b_n= L .

n→ ∞

The strategy is to find simple sequences for { a_n } and { c_n } that bound b_n

Return to Calculus 2 Notes Involving Sequences, Infinite Series, etc