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الكلية كلية التربية للعلوم الصرفة     القسم قسم الفيزياء     المرحلة 2
أستاذ المادة هدى عامر هادي       21/03/2021 20:32:35
Sequence
{1, 2, 3, 4, …}
1st term
2nd term 3rd term three dots means goes on forever (infinite)
4th term
Infinite or Finite: when the sequence goes on forever it is called an infinite sequence, otherwise it is a finite sequence.
Ex: 1) {1, 3, 5, 7, …} it is infinite sequence.
2) {20, 25, 30, 35, …} it is infinite sequence.
3) {2, 4, 6, 8} it is finite sequence.
4) {a, b, c, d, e} it is finite sequence.

Infinite Sequences and Series:
Sequence: A sequence is a list of numbers written in a definite order :-
* +.
The Sequence have, is called the first term, the second term, is the nth term.
The sequence * + is denoted by * + or * +
, some seq. can be dfind by
given a formula for nth term as shown that:-
1- *


+
*











+.
2- *
( )
( )


+
*












( )
( )


+.
3- * (


)+
*
?

(


) +.
4- *? +
* ? ? ? +



Def:The sequence * +
converges to the number , if for every positive number
there corresponds an integer N such that for all n.
| | .
If no such number exists, we say that * + diverges. If * + converges to , we
write or simply , and called the limit of the sequence.
Ex: Show that: 1-


2- .


Sol: 1) Let be given, we must show that s.t |


|


or


. If N is any integer greater than

, will hold . Then



.
2) Let be given, | | , since , this all ready it’s hold
. Then .
Ex: Check the convergence of the following sequences.
1- *


( )

+ 2- *



(


)+ 3- *


+.
Sol:
1)











, the sequence convergence to 1.
2)



(


), let u=1/n, where ,







( )



( )




. The sequence converges to

.
3)


, by using L’Hopital’s Rule =



. The Seq. Conv. To .
Theorem: Let * + and * + be sequences of real numbers and let A and B be real
numbers. The following rules hold if and .
1- Sum Rule: ( ) .
2- Difference Rule: ( ) .
3- Constant Multiple Rule: ( ) , .
4- Product Rule: ( ) .
5- Quotient Rule:





.
Theorem: (The Sandwich theorem for Sequences)
Let * + ,* + and * + be sequences of real numbers. If hold for all
n beyond some index N, and if and , then
also


Theorem: Let * + be sequence of real numbers. If and if is a function; that is
continuous at and defined at all , then ( ) ( ).
Ex: Show that ?


.
Sol: We know that

, taking ( ) ? & L=1 in theorem gives ?


?
? *

+ , ( )

,


.
Theorem: Suppose that ( ) is a function defined , and that * + is a
sequence of real numbers s.t ( ) .
Then ( ) .
Ex: Show that



The function

is defined


by L’Hopital’sRule






.



.
Ex: Does the Sequence whose nth term is converge? If so, find .


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