10.2 SequencesMath 6BCalculus II

Limit of Sequences from Limits of Functions

• Suppose f is a function such that for all positive integers n. If , then the limit of the sequence is also L.

Limit Laws for Sequences

• Assume that the sequences and have limits A and B, respectively. Then,

• , where c is a real number

• , provided

Monotonic Sequence

• A sequence {an} is called increasing if an< an+1 for all , that is, a1< a2< a3<…. It is called decreasing if an>an+1. It is called monotonic if it is either increasing or decreasing.

Bounded Sequence

• A sequence {an} is bounded above if there is a number M such that

• It is bounded below if there is a number m such that

• If it is bounded above and below, then {an} is a bounded sequence.

for all 1na M n

for all 1nm a n

Geometric Sequence

• The geometric sequence {r n} will converge if , otherwise the sequence diverges.

Limit Laws and Squeeze Theorem• Squeeze Theorem for Sequences

• Theorem

0If for and lim lim

then limn n n n n n n

n n

a b c n n a c

b L

If lim 0 then lim 0n n n na a

Monotonic Sequence Theorem

• Every bounded, monotonic sequence is convergent.

Limit of a Sequence

• The sequence converges to L provided the terms of can be made arbitrarily close to L by taking n sufficiently large. More precisely, has the unique limit L if given any tolerance , it is possible to find a positive integer N (depending on ) such that where n > N.

Limit of a Sequence

• If the limit of a sequence is L, we say the sequence converges to L, written

• A sequence that does not converge is said to diverge.