Class Notes (1,100,000)

CA (620,000)

UTSC (30,000)

Mathematics (1,000)

MATA37H3 (100)

Smith( G) (20)

Lecture 14

Department

MathematicsCourse Code

MATA37H3Professor

Smith( G)Lecture

14This

**preview**shows half of the first page. to view the full**2 pages of the document.**MATA37 - Lecture 14 - Sequences

Sequences

• See section 7.1 and 7.2

• Idea: An inﬁnite sequence of real numbers is an inﬁnite order list of real numbers.

• e.g. 1,1

2,1

3,1

4,..., 1

n, . . .

• In general, a1, a2, . . . , an, . . .

• An inﬁnite sequence is a real-valued function whose domain is N, denoted in one of the

following ways (see page 579):

–{an}∞

n=1

–{an}(no index listed means start at 1)

–an=f(n)

• The term anis called the general term.

• Examples of sequences:

–{n}

1,2,3,4, . . .

–1 + 1

2n∞

n=1

a1= 1 + 1

2, a2= 1 + 1

22, a3= 1 + 1

23

–an= 1 + (−1)n

0,2,0,2, . . .

• Let l∈R(lrepresenting the limit of {an}). We say that the sqeuence {an}converges to l

(symbolically, an→las n→ ∞, or lim

n→∞

an=l), provided ∃l∈R,∀ > 0,∃N > 0such that

∀n∈N, if n>N, then |an−l|<

• If {an}doesn’t converge, we say it diverges (see page 579).

• Example 1: Show 1

√nconverges to 0.

–Given an arbitrary > 0, we must choose N=>0, such that if n > N , then

1

√n−0

<

–Let > 0be arbitrary. Choose N=>0. Suppose n>N

–Consider

1

√n−0

. Want a multiple of some power of N.

–=

1

√n

=1

√n

1

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