高等数学英文版课件PPT 09 INFINITE SEQUENCES AND SERIES.ppt

① by using the preceding notation, ② by using the defining formula, ③ by writing out the terms of the sequence.
Notice that n doesn’t have to start at 1.
f1 1,
f2 1, fn fn1 fn2 ,
n 3.
Each term is the sum of the two preceding terms. The first terms are
1, 1, 2, 3, 5, 8, 13, 21
This sequence arose when the 13th-century Italian mathematician known as Fibonacci solved a problem concerning the breeding of rabbits (see Exercise 63).
(1) Definition A sequence has the limit L and we write
an= L or an→L as n→∞
if for every >0 there is a corresponding integer N such
that
| an-L |< , whenever n>N.
(1, a1), (2, a2), (3, a3), …, (n, an), …
From the sequence an=n/(n+1) are approaching 1 as n becomes large. In fact, the difference
1– n = 1 n 1 n 1
can be made as small as we like by taking n sufficiently large.
limx→∞f(x)=L, we see that the only difference between limn→∞an=L and lim x→∞f(x)=L is that n is required to be an integer. Thus we have the following theorem:
If
lim
n
an
exists,
we
say
the
sequence
converges
(or
is
convergent). Otherwise, we say the sequence diverges
(or is divergent).
Comparison of Definition 1 and the definition of
Theorem If lim x→∞f(x)=L and f(n)= an, where n is
an integer, then limn→∞an=L.
In particular, since we know that lim x→∞(1/xr)=0 when r>0
(Theorem 3.5.4),
NOTATION The sequence {a1, a2, a3, …} is also denoted by
an
or
a n n1
Example 1 Some sequences can be defined by giving a formula for the n-th term. In the following examples we give three descriptions of the sequence
We indicate this by writing
lim
n
n n 1
=1
In general, the notation
lim
n
an
=
L
means that the terms of the sequence {an} can be made arbitrarily close to L by taking n sufficiently large.
we have
(3)
lim
n
1 nr
0
if r 0
The analogue of Definition 3.5.7 is the following:
Chapter 9
INFINITE SEQUENCES AND SERIES
10.1 SEQUENCES
A sequence can be thought of as a list of numbers written in a definite order:
a1, a2, a3, …, an,… The number a1 is called the first term, a2 is the second term, and in general an is the n-th term. We will deal exclusively with infinite sequences and so each term will have a successor an+1 .
A sequence can be pictured either by plotting its terms on a number line or by plotting its graph in a plan
Note that, since a sequence is a function whose domain is the set of positive integers, its graph consists of isolated points with coordinate
For example:
nn ຫໍສະໝຸດ 1n1,ann n 1
1
2
,
2 3
,
3 4
,
4 5
,,
n
n
1
EXAMPLE 2 Here are some sequences that do not have a simple defining equation.
7, 1, 8, 2, 9, 1, 8, 2, 8, 4, 5,