Mechanical Engineering Dep.
Wakkas Ali
2013-2014
Engineering college
Kufa university
Infinite Series
Sequence: - A sequence {
} is a function whose domain is the set of
natural numbers.
Where an is general term (general element).
Example {
}
{
The sequence {
}
}
converges to 1.
Convergence of sequence
A sequence {
} converges to L means that
where L is a
finite single number.
Otherwise sequence is diverging.
Example {(
{(
) } is convergent sequence because
) }
Series
A series is the sum of sequence terms denoted by
∑
∑
The series ∑
( ) converges to
Convergence of series ∑
converges to S means that
1
Mathematics for 2nd stage
Infinite Series
Sn is the partial sum and S is the total sum.
Example-1) Use the definition to determine whether the series is convergent
or divergent.
∑
(
∑(
)
Solution a
(
(
)
(
(
(
(
(
(
(
(
(
(
(
(
(
(
Hence the series is divergent.
Solution b
∑(
{
Hence the series is divergent.
2
Mechanical Engineering Dep.
Wakkas Ali
Engineering college
Kufa university
2013-2014
Maclaurin’s series
If f(x) is a function defined& differentiable at x=0 then f(x) can be written in
series form as follows
(
(̅
(̅
(
(̅
(
Maclaurin’s series form for f(x)
Note n factorial n! = n*(n-1)*(n-2)* (n-3)* …….*3*2*1
Example-2) Find Maclaurin’s series (
Solution
(
(
(̅
(̅
(̅
̅(
(̅̅
(̅̅
(
(̅
(
(̅
(̅
(
∑
Example-3) Find Maclaurin’s series
(
Solution
(
(̅
(
(
)
(̅
(
)
3
Mathematics for 2nd stage
Infinite Series
(̅
(
)
(̅
(
)
(̅
(
)
(̅
(
)
(
(
(̅
(̅
(̅
(
(
∑
Taylor’s series
If f(x) is a function defined & differentiable at x=a then f(x) can be written
in series form as follows
(
(̅
(
(
(̅
(
̅
(̅
(
(
(
Taylor’s expansion of f(x) about x=a
If a=0 then Taylor’s series yields Maclaurin’s series.
Example-4) Use Taylor’s series to find the approximate for cos(620)
Solution
Let (
(
(̅
4
(
(
(
(
(̅
(
(
√
Mechanical Engineering Dep.
Wakkas Ali
(̅
(̅
(
(
(
(̅
Engineering college
Kufa university
2013-2014
(
(̅
(
(
√
(
(
)
Example-5) Find
(
(
)
using series method
Solution
(
(
(
(
(
5
Mathematics for 2nd stage
Example-6) Find ∫
Infinite Series
using series method
Solution
∫
∫(
(
(
)
]
(
Geometric series
It’s a series of the form
∑
This series is divergent if -1≥ x ≥1
This series is convergent if -1< x <1 and it’s sum is
Or
Example-7) Find the sum of the following series
∑
∑
(
Solution a
∑
∑
This is a geometric series
6
∑(
)
∑(
)
Mechanical Engineering Dep.
Wakkas Ali
[
]
Engineering college
Kufa university
2013-2014
*
+
Solution b
∑
(
(
∑
(
∑(
)
This is a geometric series
[
]
*
+
Tests for convergence:then ∑
1. The nth general term test:- if
2. Comparison test:- a) if an ≤ bn and ∑
is divergent.
convergent then ∑
is
convergent.
b) if an ≥ bn and ∑
divergent then ∑
is divergent.
3. The integral test:- If f(x) is a decreasing continuous positive function
for
then ∑ (
(
and ∫
converges or diverges together.
then ∑
4. The limit comparison test:- If
and ∑
converges or diverges together.
5. Limit ratio test:- If
∑
∑
6. the limit nth root test: If
7
Mathematics for 2nd stage
Infinite Series
∑
√
√
∑
Example-8) Test the following series for convergence ∑
(
)
Solution
(
)
Example-9) Check ∑
( ) for convergence using integral method.
Solution
Let (
F(x) is continuous decreasing function.
∫ (
∫
]
]
(
)
∑
Theorem (P- series)∑
Example-10) Test ∑
converges for P >1 and diverges for P ≤ 1
for convergence.
Solution
Let
(
∫ (
8
x≥2
∫
( (
]
Mechanical Engineering Dep.
Wakkas Ali
( (
2013-2014
]
Engineering college
Kufa university
( (
∑
Example-11) Test ∑
for convergence.
Solution
Since
But ∑
then
is convergent p-series p=5 > 1
∑
Example-12) Test ∑
for convergence.
Solution
Let ∑
∑
Let ∑
∑
convergent p-series p > 1
∑
Example-13) Test ∑
for convergence.
Solution
(
9
Mathematics for 2nd stage
Infinite Series
(
(
)
∑
Example-14) Test ∑
for convergence.
Solution
(
(
(
(
(
(
(
(
(
)
(
(
)
∑
Example-15) Test ∑
(
)
for convergence.
Solution
√(
√
∑
(
)
(
)
)
Alternating series:- it’s a series of general form
∑(
10
Mechanical Engineering Dep.
Wakkas Ali
Engineering college
Kufa university
2013-2014
Convergence of alternating series
i.
If ∑
convergent
∑(
ii.
If ∑
divergent &
converges absolutely.
∑(
converges
∑(
diverges.
conditionally.
If ∑
iii.
divergent &
Power series (series of functions)
It’s a series of general form
∑
Interval of convergence
It’s the value of x that makes power series converges absolutely and to find
it any convergence test may be used.
Example-16) find interval of convergence of
∑
(
∑
(
√
Solution a)
(
(
(
|
|
|
|
|
(
(
(
(
|
(
|
|
|
|
11
Mathematics for 2nd stage
Infinite Series
Solution a)
(
(
√
|
√
|
|
|
|
12
|
(
√
√
|
(
√
|
|
|