Chapter 1 Infinite Series
Generally, for the given sequence
,.......,
......,3,21n a a a a the
expression formed by the sequence ,.......,......,3,21n a a a a .......,.....321+++++n a a a a
is called the infinite series of the constants term, denoted by ∑∞
=1n n a , that is
∑∞
=1
n n a =.......,.....321+++++n a a a a
Where the nth term is said to be the general term of the series, moreover, the nth partial sum of the series is given by
=n S ......321n a a a a ++++
Determine whether the infinite series converges or diverges.
Whil e it’s possible to add two numbers, three numbers, a hundred numbers, or even a million numbers, it’s impossible to add an infinite number of numbers.
To form an infinite series we begin with an infinite sequence of real numbers: .....,,,3210a a a a , we can not form the sum of all the k a (there is an infinite number of the term), but we can form the partial sums
∑===0
000k k a a S
∑==+=1
101k k a a a S
∑==++=2
2102k k a a a a S
∑==+++=3
32103k k a a a a a S
……………….
∑==+++++=n
k k n n a a a a a a S 03210.......
Definition 1.1.1
If the sequence {n S } of partial sums has a finite limit L, We write ∑∞
==0k k a L
and say that the series ∑∞
=0
k k a converges to L. we call L the
sum of the series.
If the limit of the sequence {n S } of partial sums don’t exists, we say that the series ∑∞
=0k k a diverges.
Remark it is important to note that the sum of a series is not a sum in the ordering sense. It is a limit.
EX 1.1.1 prove the following proposition: Proposition1.1.1:
(1) If 1=0
k k a converges, and ;110
x
x k k -=
∑∞
= (2)If ,1≥x then the ∑∞
=0
k k x diverges.
Proof: the nth partial sum of the geometric series ∑∞
=0
k k a
takes the form 1321.......1-+++++=n n x x x x S ① Multiplication by x gives
).......1(1321-+++++=n n x x x x x xS =n n x x x x x +++++-1321.......
Subtracting the second equation from the first, we find that
n n x S x -=-1)1(. For ,1≠x this gives
x
x S n
n --=11 ③
If ,1x x x S n n n n -=--=→→11
11lim lim 0
0 This proves (1).
Now let us prove (2). For x=1, we use equation ① and device that ,n S n =
Obviously, ∞=∞
→n n S lim , ∑∞
=0k k a diverges.
For x=-1 we use equation ① and we deduce If n is odd, then 0=n S , If n is even, then .1-=n S The
sequence
of
partial
sum
n
S like this
0,-1,0,-1,0,-1………..
Because the limit of sequence }{n S of partial sum does not exist.
