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and thus
sin(x) = x ¡
1
3!
x
3
+
1
5!
x
5
¡···:
Function cos(x) has the following series expansion
cos(x) = 1 ¡
1
2!
x
2
+
1
4!
x
4
¡···:
The domain of convergence of sine an d cosine functions are 1.
4.1.3 Partial sums and convergence
Obviously, we are unable to add up infinite terms of a series directly and calculate its
value. Therefore, we should consider the partial sums of an infinite series
S
n
(x) =
X
k=1
n
c
k
(x ¡x
0
)
k
:
Therefore, we find a function sequence (S
n
(x)) for n = 0 ; 1; ···, that we can study its con-
vergence. In addition to the pointwise convergence of the sequence, that is,
lim
n!1
S
n
(a) = f (a);
for any a in the domain of convergence of f, we can define the notion of uniform conver-
gence of the sequence. Let us first see an example.
Example 4.5. Consider f unction f(x) =
1
1 ¡ x
2
in (¡1 ; 1) with the series expansion
1
1 ¡x
2
= 1 + x
2
+ x
4
+ x
6
+ ···:
Now, consider the following partial sum
S
n
(x) = 1 + x
2
+ x
4
+ ·· ·+ x
2n
=
1 ¡x
2n+2
1 ¡x
2
:
Note that if jxj < 1 then x
2n+2
n!1
0. Fig.4.1 shows the graph of function f (x) and
S
n
(x) for a few values of n. Also note that f(x) goes unbounded at x = ±1 and no partial
sums (of any terms) is able to catch up the function in adjacent of these two points. In
other words, for any n > 0, there is some point x
n
2(¡1; 1) such that
jf (x
n
) ¡S
n
(x
n
)j> 1:
On the other hand, let us restrict the domain to
¡1 +
1
m
; 1 ¡
1
m
for any m > 1. Then for
any " > 0, there is n such that
max
x2
h
¡1+
1
m
;1¡
1
m
i
jf(x) ¡S
n
(x)j< ":
For example, for [¡0.99; 0.99] and " = 0.1, we can choose as large as n = 310 to make sure
the above inequality holds for all x in the closed interval .
6 Series Solution of Linear Equations