Diziler

Diziler

Bir dizi, bazı sayıların belirli bir sıraya gore dizilmesi olarak dusunulebilir.

a1, a2, · · · , an, · · ·

Ornegin,2, 4, 6, 8, 10, . . . , 2n, . . .

seklindeki bir dizide, a1 = 2 ilk terim, a2 = 4 ikinci terim ve genelolarak an = 2n ise n. terim olarak adlandırılır.

Diziler

Biz genellikle sonsuz dizilerle ilgilenecegiz.Dolayısıyla, her an teriminden sonra gelen bir an+1 terimi olacaktır.

Her pozitif n dogal sayısı icin dizide bir an terimi vardır. Bu sekilde, birdiziyi tanım kumesi dogal sayılar (N+) olan bir fonksiyon olaraktanımlayabiliriz.Ancak, fonksiyonun n’de aldıgı degeri gostermek icin f(n) yerine anyazacagız.

{a1, a2, · · · , an, · · · } dizisi {an} veya {an}∞n=1 ile de gosterilir.

Diziler

Ornek 1Asagıdaki orneklerin her birindeki dizi uc farklı gosterim ile verilmistir.Bunlardan birincisinde onceki gosterim, ikincisinde genel terimkullanılmıstır. Ucuncusunde ise dizi, terimleri tek tek yazılarak verilmistir.{

n

n+ 1

}∞

n=1

an =n

n+ 1

{1

2,2

3,3

4, · · · , n

n+ 1, · · ·

}{(−1)n n

3n

}∞

n=1an = (−1)n n

3n

{−1

3,2

9,− 3

27, · · · , (−1)n n

3n, · · ·

}{√

n− 3}∞n=3

an =√n− 3, n ≥ 3

{0, 1,√2, · · · ,

√n− 3, · · ·

}{cos(nπ6

)}∞

n=0an = cos

(nπ6

), n ≥ 0

{1,

√3

2,1

2, · · · , cos

(nπ6

), · · ·

}

Bir dizide n’nin 1’den baslamak zorunda olmadıgına dikkat ediniz.

Diziler

an =n

n+ 1dizisi gibi herhangi bir dizi, Sekildeki gibi sayı dogrusu

uzerinde veya grafigi cizilerek gosterilebilir.

EXAMPLE 2 Here are some sequences that don’t have a simple defining equation.(a) The sequence , where is the population of the world as of January 1 inthe year .(b) If we let be the digit in the decimal place of the number , then is awell-defined sequence whose first few terms are

(c) The Fibonacci sequence is defined recursively by the conditions

Each term is the sum of the two preceding terms. The first few terms are

This sequence arose when the 13th-century Italian mathematician known asFibonacci solved a problem concerning the breeding of rabbits (see Exercise 37).

A sequence such as the one in Example 1(a), , can be picturedeither by plotting its terms on a number line, as in Figure 1, or by plotting its graph,as in Figure 2. Note that, since a sequence is a function whose domain is the set ofpositive integers, its graph consists of isolated points with coordinates

. . . . . .

From Figure 1 or 2 it appears that the terms of the sequence areapproaching 1 as becomes large. In fact, the difference

can be made as small as we like by taking sufficiently large. We indicate this by writing

In general, the notation

means that the terms of the sequence approach as becomes large. Notice thatthe following definition of the limit of a sequence is very similar to the definition of alimit of a function at infinity given in Section 2.5.

nL�an �

limn l �

an � L

limn l �

n

n � 1� 1

n

1 �n

n � 1�

1

n � 1

nan � n��n � 1�

FIGURE 2

0 n

an

1

1

2 3 4 5 6 7

78a¶=

0 112

a¡ a™ a£a¢

FIGURE 1

�n, an ��3, a3 ��2, a2 ��1, a1�

an � n��n � 1�

�1, 1, 2, 3, 5, 8, 13, 21, . . .�

n � 3fn � fn�1 � fn�2f2 � 1f1 � 1

� fn �

�7, 1, 8, 2, 8, 1, 8, 2, 8, 4, 5, . . .�

�an �enthan

npn�pn �

564 � CHAPTER 8 INFINITE SEQUENCES AND SERIES

EXAMPLE 2 Here are some sequences that don’t have a simple defining equation.(a) The sequence , where is the population of the world as of January 1 inthe year .(b) If we let be the digit in the decimal place of the number , then is awell-defined sequence whose first few terms are

(c) The Fibonacci sequence is defined recursively by the conditions

Each term is the sum of the two preceding terms. The first few terms are

This sequence arose when the 13th-century Italian mathematician known asFibonacci solved a problem concerning the breeding of rabbits (see Exercise 37).

A sequence such as the one in Example 1(a), , can be picturedeither by plotting its terms on a number line, as in Figure 1, or by plotting its graph,as in Figure 2. Note that, since a sequence is a function whose domain is the set ofpositive integers, its graph consists of isolated points with coordinates

. . . . . .

From Figure 1 or 2 it appears that the terms of the sequence areapproaching 1 as becomes large. In fact, the difference

can be made as small as we like by taking sufficiently large. We indicate this by writing

In general, the notation

means that the terms of the sequence approach as becomes large. Notice thatthe following definition of the limit of a sequence is very similar to the definition of alimit of a function at infinity given in Section 2.5.

nL�an �

limn l �

an � L

limn l �

n

n � 1� 1

n

1 �n

n � 1�

1

n � 1

nan � n��n � 1�

FIGURE 2

0 n

an

1

1

2 3 4 5 6 7

78a¶=

0 112

a¡ a™ a£a¢

FIGURE 1

�n, an ��3, a3 ��2, a2 ��1, a1�

an � n��n � 1�

�1, 1, 2, 3, 5, 8, 13, 21, . . .�

n � 3fn � fn�1 � fn�2f2 � 1f1 � 1

� fn �

�7, 1, 8, 2, 8, 1, 8, 2, 8, 4, 5, . . .�

�an �enthan

npn�pn �

564 � CHAPTER 8 INFINITE SEQUENCES AND SERIES

Sekilden de anlasılabilecegi gibi, n sayısı buyudukce an =n

n+ 1dizisinin

terimleri 1 e yaklasır.

Diziler

Gercekten de, n sayısı yeteri kadar buyuk alınarak

1− n

n+ 1=

1

n+ 1

farkı istenildigi kadar kucuk yapılabilir. Biz bunu

limn→∞

n

n+ 1= 1

yazarak ifade ediyoruz.

Genel olarak,limn→∞

an = L

gosterimi, n sayısı buyudukce {an} dizisinin terimlerinin L ye yaklastıgınıifade etmek icin kullanılır.

Diziler

Tanım 2n sayısı yeteri kadar buyuk secilerek an terimleri L’ye istenildigi kadaryakın yapılabiliyorsa {an} dizisinin limiti L’dir ve

limn→∞

an = L ya da n→∞ icin an → L

yazılır. Eger limn→∞

an degeri varsa {an} dizi yakınsaktır. Aksi durumda;

dizi, ıraksaktır.

Asagıdaki Sekilde limn→∞

an = L olan iki dizinin grafikleri gosterilmistir.

Definition A sequence has the limit and we write

if we can make the terms as close to as we like by taking sufficientlylarge. If exists, we say the sequence converges (or is convergent).Otherwise, we say the sequence diverges (or is divergent).

Figure 3 illustrates Definition 1 by showing the graphs of two sequences that havethe limit .

If you compare Definition 1 with Definition 2.5.4 you will see that the only differ-ence between and is that is required to be an inte-ger. Thus, we have the following theorem, which is illustrated by Figure 4.

Theorem If and when is an integer, then .

In particular, since we know from Section 2.5 that when we have

if

If becomes large as n becomes large, we use the notation

In this case the sequence is divergent, but in a special way. We say that di-verges to .

The Limit Laws given in Section 2.3 also hold for the limits of sequences and theirproofs are similar.

��an ��an �

limn l �

an � �

an

r � 0limn l �

1

nr � 03

r � 0,limx l � �1�xr� � 0

FIGURE 4 20 x

y

1 3 4

L

y=ƒ

limn l � an � Lnf �n� � anlimx l � f �x� � L2

nlimx l � f �x� � Llimn l � an � L

0 n

an

L

0 n

an

LFIGURE 3Graphs of twosequences withlim an= Ln  `

L

limn l � an

nLan

an l L as n l �orlimn l �

an � L

L�an �1

SECTION 8.1 SEQUENCES � 565

� A more precise definition of the limitof a sequence is given in Appendix D.

Definition A sequence has the limit and we write

if we can make the terms as close to as we like by taking sufficientlylarge. If exists, we say the sequence converges (or is convergent).Otherwise, we say the sequence diverges (or is divergent).

Figure 3 illustrates Definition 1 by showing the graphs of two sequences that havethe limit .

If you compare Definition 1 with Definition 2.5.4 you will see that the only differ-ence between and is that is required to be an inte-ger. Thus, we have the following theorem, which is illustrated by Figure 4.

Theorem If and when is an integer, then .

In particular, since we know from Section 2.5 that when we have

if

If becomes large as n becomes large, we use the notation

In this case the sequence is divergent, but in a special way. We say that di-verges to .

The Limit Laws given in Section 2.3 also hold for the limits of sequences and theirproofs are similar.

��an ��an �

limn l �

an � �

an

r � 0limn l �

1

nr � 03

r � 0,limx l � �1�xr� � 0

FIGURE 4 20 x

y

1 3 4

L

y=ƒ

limn l � an � Lnf �n� � anlimx l � f �x� � L2

nlimx l � f �x� � Llimn l � an � L

0 n

an

L

0 n

an

LFIGURE 3Graphs of twosequences withlim an= Ln  `

L

limn l � an

nLan

an l L as n l �orlimn l �

an � L

L�an �1

SECTION 8.1 SEQUENCES � 565

� A more precise definition of the limitof a sequence is given in Appendix D.

Diziler

Bu Tanım ve fonksiyonların limiti tanımı karsılastırıldıgında gorulecegigibi lim

n→∞an = L ve lim

x→∞f(x) = L arasındaki tek fark, ilk limitte n nin

dogal sayı olmasıdır.

Definition A sequence has the limit and we write

if we can make the terms as close to as we like by taking sufficientlylarge. If exists, we say the sequence converges (or is convergent).Otherwise, we say the sequence diverges (or is divergent).

Figure 3 illustrates Definition 1 by showing the graphs of two sequences that havethe limit .

If you compare Definition 1 with Definition 2.5.4 you will see that the only differ-ence between and is that is required to be an inte-ger. Thus, we have the following theorem, which is illustrated by Figure 4.

Theorem If and when is an integer, then .

In particular, since we know from Section 2.5 that when we have

if

If becomes large as n becomes large, we use the notation

In this case the sequence is divergent, but in a special way. We say that di-verges to .

The Limit Laws given in Section 2.3 also hold for the limits of sequences and theirproofs are similar.

��an ��an �

limn l �

an � �

an

r � 0limn l �

1

nr � 03

r � 0,limx l � �1�xr� � 0

FIGURE 4 20 x

y

1 3 4

L

y=ƒ

limn l � an � Lnf �n� � anlimx l � f �x� � L2

nlimx l � f �x� � Llimn l � an � L

0 n

an

L

0 n

an

LFIGURE 3Graphs of twosequences withlim an= Ln  `

L

limn l � an

nLan

an l L as n l �orlimn l �

an � L

L�an �1

SECTION 8.1 SEQUENCES � 565

� A more precise definition of the limitof a sequence is given in Appendix D.

Teorem 3limx→∞

f(x) = L ve her n dogal sayısı icin f(n) = an ise limn→∞

an = L olur.

Diziler

Ozel olarak, r > 0 icin limx→∞

1

xr= 0 oldugu bilindiginden, r > 0 icin

limn→∞

1

nr= 0

yazılır.

Buyuk n degerleri icin an de buyuk degerler alıyorsa,

limn→∞

an =∞

yazılır. Bu durumda {an} dizisi ıraksaktır. Ancak bu ozel ıraksak olmadurumunu diger ıraksaklıklardan ayırarak, {an} dizisi sonsuza ıraksardiyecegiz.

Diziler

Yakınsak Dizilerde Limit Kuralları{an} ve {bn} iki yakınsak dizi ve c bir sabit olmak uzere asagıdakilersaglanır.

• limn→∞

(an + bn) = limn→∞

an + limn→∞

bn

• limn→∞

(an − bn) = limn→∞

an − limn→∞

bn

• limn→∞

(can) = c limn→∞

an

• limn→∞

(anbn) = limn→∞

an · limn→∞

bn

• limn→∞

bn 6= 0 ise limn→∞

anbn

=lim

n→∞an

limn→∞

bn

• p > 0 ve an > 0 ise limn→∞

apn =[limn→∞

an]p

Diziler

If and are convergent sequences and is a constant, then

The Squeeze Theorem can also be adapted for sequences as follows (see Figure 5).

If for and , then .

Another useful fact about limits of sequences is given by the following theorem,which follows from the Squeeze Theorem because .

Theorem If , then .

EXAMPLE 3 Find .

SOLUTION The method is similar to the one we used in Section 2.5: Divide numeratorand denominator by the highest power of that occurs in the denominator and thenuse the Limit Laws.

Here we used Equation 3 with .

EXAMPLE 4 Calculate .limn l �

ln n

n

r � 1

�1

1 � 0� 1

limn l �

n

n � 1� lim

n l �

1

1 �1

n

�limn l �

1

limn l �

1 � limn l �

1

n

n

limn l �

n

n � 1

limn l �

an � 0limn l �

� an � � 04

�� an � � an � � an �

limn l �

bn � Llimn l �

an � limn l �

cn � Ln � n0an � bn � cn

limn l �

anp � [lim

n l � an]p if p � 0 and an � 0

limn l �

an

bn�

lim n l �

an

limn l �

bnif lim

n l � bn � 0

limn l �

�an bn � � limn l �

an � limn l �

bn

limn l �

c � c limn l �

can � c limn l �

an

limn l �

�an � bn � � limn l �

an � limn l �

bn

limn l �

�an � bn � � limn l �

an � limn l �

bn

c�bn ��an �

566 � CHAPTER 8 INFINITE SEQUENCES AND SERIES

Limit Laws for Convergent Sequences

FIGURE 5The sequence �bn� is squeezed betweenthe sequences �an� and �cn�.

0 n

cn

an

bn

Squeeze Theorem for Sequences

� This shows that the guess we madeearlier from Figures 1 and 2 was correct.

Sıkıstırma teoremi diziler icin asagıdaki sekilde uyarlanabilir.

Teorem 4n ≥ n0 icin an ≤ bn ≤ cn saglansın.

limn→∞

an = limn→∞

cn = L ise limn→∞

bn = L olur.

Diziler

Diziler hakkındaki diger yararlı bir sonuc olan asagıdaki teorem,−|an| ≤ an ≤ |an| oldugundan Sıkıstırma Teoreminden elde edilir.

Teorem 5limn→∞

|an| = 0 ise limn→∞

an = 0 olur.

Diziler

Ornek 6limn→∞

n

n+ 1limitini bulunuz.

Cozum.

Kesrin pay ve paydasını, n nin paydada gorulen en buyuk kuvvetine bolerve limit kurallarını kullanırsak

limn→∞

n

n+ 1= lim

n→∞��>

1n

�n(1 + 1

n

)=

limn→∞

1

limn→∞

1 + limn→∞

1n

=1

1 + 0= 1

sonucuna ulasırız.

Diziler

Ornek 7

limn→∞

lnn

nlimitini hesaplayınız.

Cozum.

Burada, n→∞ iken hem pay hem de pay da sonsuza gitmektedir.L’Hospital kuralını dogrudan uygulayamayız cunku bu kural dizilere degilgercel degerli fonksiyonlara uygulanabilmektedir. Ancak, L’Hospitalkuralını bu dizi ile cok yakından ilgili olan f(x) = lnx

x fonksiyonunauygulayabilir ve

limx→∞

lnx

x= lim

x→∞

1/x

1= 0

buluruz. Boylece

limn→∞

lnn

n= 0

elde ederiz.

Diziler

Ornek 8{an} = {(−1)n} dizisinin yakınsak olup olmadıgını belirleyiniz.

Cozum.

Bu dizinin terimlerini tek tek yazarsak {−1, 1,−1, 1,−1, 1,−1, 1,−1, . . .}elde ederiz.

SOLUTION Notice that both numerator and denominator approach infinity as .We can’t apply l’Hospital’s Rule directly because it applies not to sequences but tofunctions of a real variable. However, we can apply l’Hospital’s Rule to the relatedfunction and obtain

Therefore, by Theorem 2 we have

EXAMPLE 5 Determine whether the sequence is convergent or divergent.

SOLUTION If we write out the terms of the sequence, we obtain

The graph of this sequence is shown in Figure 6. Since the terms oscillate between 1and infinitely often, does not approach any number. Thus, doesnot exist; that is, the sequence is divergent.

EXAMPLE 6 Evaluate if it exists.

SOLUTION

Therefore, by Theorem 4,

EXAMPLE 7 Discuss the convergence of the sequence , where.

SOLUTION Both numerator and denominator approach infinity as but here wehave no corresponding function for use with l’Hospital’s Rule ( is not definedwhen is not an integer). Let’s write out a few terms to get a feeling for what hap-pens to as gets large:

It appears from these expressions and the graph in Figure 8 that the terms aredecreasing and perhaps approach 0. To confirm this, observe from Equation 5 that

an �1

n 2 � 3 � � � � � n

n � n � � � � � n

an �1 � 2 � 3 � � � � � n

n � n � n � � � � � n5

a3 �1 � 2 � 3

3 � 3 � 3a2 �

1 � 2

2 � 2a1 � 1

nan

xx!n l �

n! � 1 � 2 � 3 � � � � � nan � n!�nn

limn l �

��1�n

n� 0

limn l �

� ��1�n

n � � limn l �

1

n� 0

limn l �

��1�n

n

���1�n �limn l � ��1�nan�1

��1, 1, �1, 1, �1, 1, �1, . . .�

an � ��1�n

limn l �

ln n

n� 0

limx l �

ln x

x� lim

x l � 1�x

1� 0

f �x� � �ln x��x

n l �

SECTION 8.1 SEQUENCES � 567

0 n

an

1

1

2 3 4

_1

FIGURE 6

FIGURE 7

0 n

an

1

1

_1

� The graph of the sequence inExample 6 is shown in Figure 7 and supports the answer.

Terimler, (−1) ile 1 arasında devamlı gidip geldigi icin an hic bir sayıyayaklasmaz. Dolayısıyla, lim

n→∞(−1)n degeri yoktur. Baska bir deyisle,

(−1)n dizisi ıraksaktır.

Diziler

Ornek 9

{an} ={(−1)n

n

}limitini arastırınız.

Cozum.

Ilk olarak,

limn→∞

∣∣∣∣(−1)nn

∣∣∣∣ = limn→∞

1

n= 0

Elde edilir. Dolayısıyla, Teorem 5 den

limn→∞

(−1)n

n= 0

bulunur.

Diziler

Ornek 10Hangi r degerleri icin {rn} dizisi yakınsaktır?

Cozum.

a > 1 icin limx→∞

ax =∞ ve 0 < a < 1 icin limx→∞

ax = 0 olur. Simdi, a = r

alarak Teorem 4 den

limn→∞

rn =

{∞, r > 1

0, 0 < r < 1

bulunur. r = 1 ve r = 0 oldugunda ise

limn→∞

1n = limn→∞

1 = 1 ve limn→∞

0n = limn→∞

0 = 0

bulunur.

Diziler

Cozum(devamı).

Ayrıca, −1 < r < 0 ise 0 < |r| < 1 saglanır. Boylece,

limn→∞

∣∣rn∣∣ = limn→∞

|r|n = 0

olur ve bu nedenle Teorem 5 den limn→∞

rn = 0 bulunur. Eger r < −1 veya

r = −1 ise {rn} ıraksaktır. r nin cesitli degerleri icin dizinin grafigiSekilde verilmistir.

so

We know that as . Therefore, as by the Squeeze Theorem.

EXAMPLE 8 For what values of is the sequence convergent?

SOLUTION We know from Section 2.5 and the graphs of the exponential functions inSection 1.5 that for and for . There-fore, putting and using Theorem 2, we have

For the cases and we have

and

If , then , so

and therefore by Theorem 4. If , then diverges as inExample 5. Figure 9 shows the graphs for various values of . (The case isshown in Figure 6.)

The results of Example 8 are summarized for future use as follows.

The sequence is convergent if and divergent for all othervalues of .

Definition A sequence is called increasing if for all , thatis, It is called decreasing if for all . It iscalled monotonic if it is either increasing or decreasing.

n � 1an � an�1a1 a2 a3 � � � .n � 1an an�1�an �

limn l �

rn � �0

1

if �1 r 1

if r � 1

r�1 r � 1�rn�6

FIGURE 9The sequence an=rn

0 n

an

1

1

r>1

r=1

0<r<1

0 n

an

11

r<_1

_1<r<0

r � �1r�rn�r � �1limn l � rn � 0

limn l �

� rn � � limn l �

� r �n � 0

0 � r � 1�1 r 0

limn l �

0 n � limn l �

0 � 0limn l �

1n � limn l �

1 � 1

r � 0r � 1

limn l �

rn � ��

0

if r � 1

if 0 r 1

a � r0 a 1limx l � ax � 0a � 1limx l � ax � �

�rn�r

n l �an l 0n l �1�n l 0

0 an �1

n

568 � CHAPTER 8 INFINITE SEQUENCES AND SERIES

� Creating Graphs of SequencesSome computer algebra systems havespecial commands that enable us to cre-ate sequences and graph them directly.With most graphing calculators, how-ever, sequences can be graphed byusing parametric equations. For instance,the sequence in Example 7 can begraphed by entering the parametricequations

and graphing in dot mode starting with, setting the -step equal to . The

result is shown in Figure 8.1tt � 1

x � t y � t!�t t

FIGURE 8

1

0 10

so

We know that as . Therefore, as by the Squeeze Theorem.

EXAMPLE 8 For what values of is the sequence convergent?

SOLUTION We know from Section 2.5 and the graphs of the exponential functions inSection 1.5 that for and for . There-fore, putting and using Theorem 2, we have

For the cases and we have

and

If , then , so

and therefore by Theorem 4. If , then diverges as inExample 5. Figure 9 shows the graphs for various values of . (The case isshown in Figure 6.)

The results of Example 8 are summarized for future use as follows.

The sequence is convergent if and divergent for all othervalues of .

Definition A sequence is called increasing if for all , thatis, It is called decreasing if for all . It iscalled monotonic if it is either increasing or decreasing.

n � 1an � an�1a1 a2 a3 � � � .n � 1an an�1�an �

limn l �

rn � �0

1

if �1 r 1

if r � 1

r�1 r � 1�rn�6

FIGURE 9The sequence an=rn

0 n

an

1

1

r>1

r=1

0<r<1

0 n

an

11

r<_1

_1<r<0

r � �1r�rn�r � �1limn l � rn � 0

limn l �

� rn � � limn l �

� r �n � 0

0 � r � 1�1 r 0

limn l �

0 n � limn l �

0 � 0limn l �

1n � limn l �

1 � 1

r � 0r � 1

limn l �

rn � ��

0

if r � 1

if 0 r 1

a � r0 a 1limx l � ax � 0a � 1limx l � ax � �

�rn�r

n l �an l 0n l �1�n l 0

0 an �1

n

568 � CHAPTER 8 INFINITE SEQUENCES AND SERIES

� Creating Graphs of SequencesSome computer algebra systems havespecial commands that enable us to cre-ate sequences and graph them directly.With most graphing calculators, how-ever, sequences can be graphed byusing parametric equations. For instance,the sequence in Example 7 can begraphed by entering the parametricequations

and graphing in dot mode starting with, setting the -step equal to . The

result is shown in Figure 8.1tt � 1

x � t y � t!�t t

FIGURE 8

1

0 10

Diziler

{rn} dizisi −1 < r ≤ 1 icin yakınsak olup diger r degerleri icin ıraksaktır.

limn→∞

rn =

{0, −1 < r < 1

1, r = 1.

Diziler

Tanım 11Her n ≥ 1 icin an < an+1 saglanıyorsa (baska bir deyisle, a1 < a2 < . . .ise) {an} dizisine artan denir.Her n ≥ 1 icin an > an+1 saglanıyorsa {an} dizisine azalan denir.Artan veya azalan bir diziye monoton denir.

Ornek 12{3

n+5

}dizisi azalandır cunku her n ≥ 1 icin

3

n+ 5>

3

(n+ 1) + 5=

3

n+ 6

saglanır.

Diziler

Ornek 13{n

n2 + 1

}dizisinin azalan oldugunu gosteriniz.

Cozum.

f(x) =x

x2 + 1fonksiyonunu alalım. x2 > 1 oldugunda

f ′(x) =1 · (x2 + 1)− x · 2x

(x2 + 1)2=

1− x2

(x2 + 1)2< 0

olur. Dolayısıyla, f(x) fonksiyonu (1,∞) aralıgında azalandır ve boylecef(n) > f(n+ 1) dir. Baska bie deyisle {an} azalan bir dizidir.

Diziler

Tanım 14Her n ≥ 1 icin

an ≤M

saglanacak sekilde bir M sayısı varsa {an} dizisine ustten sınırlı,her n ≥ 1 icin

an ≥ m

saglanacak sekilde bir m sayısı varsa {an} dizisine alttan sınırlı denir.Hem alttan hem de ustten sınırlı olan bir diziye sınırlı dizi denir.

Ornegin, an = n dizisi alttan sınırlıdır (an > 0) ancak ustten sınırlıdegildir.

an = nn+1 dizisi sınırlıdır cunku her n ≥ 1 icin 0 < an < 1 olur.

Diziler

Teorem 15 (Monoton Dizi Teoremi)

Sınırlı ve monoton her dizi yakınsaktır.

Diziler

Ornek 16

an =2n

3n+1dizisinin yakınsaklıgını arastırınız.

Cozum.

Dizinin ilk birkac terimini yazarak baslayalım;{29 ,

427 ,

881 ,

16243 ,

32729 , . . .

}.

Buradan goruldugu gibi dizinin terimleri 0 ve 29 arasında degerler

almaktadır, dolayısıyla dizi sınırlıdır. {an} dizisinin monoton olupolmadıgını kontrol etmeliyiz.

an+1 − an =2n+1

3n+2− 2n

3n+1=

2n

3n+1

(2

3− 1

)= −1

3

2n

3n+1< 0

oldugundan dizi azalandır, dolayısıyla monotondur. Monoton DiziTeoremi nedeniyle an = 2n

3n+1 dizisinin yakınsak oldugunusoyleyebiliriz.