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Bridge of Don Academy – Department of Mathematics
Advanced Higher: Unit 3 – Sequences and Series
SEQUENCES AND SERIES
Learning Objectives:
1.
Revise knowledge gained in Unit 2 (Arithmetic & Geometric Sequences and Series).
2.
Study power series and the Maclaurin series.
3.
Use the Maclaurin series to expand ex, sinx, cosx, tanx, (1 + x)a, ln(1 + x), knowing their
range and validity.
4.
Find the Maclaurin series for simple composites of the above e.g. e2x.
5.
Use the Maclaurin series expansion to find power series for simple functions to a stated
number of terms.
6.
Use iterative schemes of the form xn+1 = g(xn), n = 0, 1, 2 ….. to solve equations where
x = g(x) is a rearrangement of the original equation.
7.
Use graphical techniques to locate an appropriate solution x0.
8.
Know the condition for convergence of the sequence {xn} given by xn+1 = g(xn), n = 0, 1.....
Remember, in Unit 2 we studied:
1.
2.
Arithmetic Sequences & Series
common difference
d = un+1 – un
nth term formula
un = a + (n – 1)d
sum to n terms of an arithmetic series
Sn  n  2a  (n  1)d ) 
2
Geometric Sequences & Series
common ratio
sum to n terms of a geometric series
sum to infinity for a geometric series where –1 < r < 1
un 1
un
a 1  r n 
Sn 
1 r
S  a
1 r
r
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Bridge of Don Academy – Department of Mathematics
Advanced Higher: Unit 3 – Sequences and Series
POWER SERIES
f ( x)  a  bx  cx 2  dx 3
Let us consider
 f ( x) 
 f ( x) 

f (0)  a
b  2cx  3dx 2

2c  6dx

f (0)  b
f (0)  2c

f (0)  6d .
 f ( x) 
6d
f ( x) 
a  bx  cx 2

dx 3
f (0)
f (0) 2 f (0) 3
 f (0) 
x
x 
x
1!
2!
3!
Hence,
Any series of the form
a0  a1 x  a2 x 2  a3 x 3  .....  ar x r  .....
is called a POWER SERIES.
Some get bigger and bigger i.e. DIVERGE
others get smaller and smaller i.e. CONVERGE.
MACLAURIN’S THEOREM
Maclaurin’s Theorem states that under certain circumstances any function can be expressed as a
power series.

1
2
3
4
n
f ( x)  f (0)  x f (0)  x f (0)  x f (0)  x f (4) (0)  .....  x f (n) (0)
1!
2!
3!
4!
n!
This is true if f(n)(0) exists for all values of n.
Examples:
1.
Express f(x) = ex as a Maclaurin’s expansion.
f ( x)  e x

f (0)  eo  1
f ( x)  e x

f (0)  eo  1
f ( x)  e x

f (0)  eo  1
f ( x)  e x

f (0)  eo  1
f ( x)  e x

f (0)  eo  1
and so on, to give
1
2
3
f ( x )  f (0)  x f (0)  x f (0)  x f (0)  .....
1!
2!
3!
2
3
 1  x  1  x  1  x  1  .....
1!
2!
3!
2
3
4
 1  x  x  x  x  .....
1! 2! 3! 4!
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Bridge of Don Academy – Department of Mathematics
2.
Advanced Higher: Unit 3 – Sequences and Series
f(x) = sin x as a Maclaurin’s expansion.
Express
f ( x)  sin x

f (0) 
f ( x) 

f (0) 
f ( x) 

f (0) 
f ( x) 

f (0) 
1
2
3
 f ( x )  f (0)  x f (0)  x f (0)  x f (0)  .....
1!
2!
3!

3.
Express
f(x) = cos x as a Maclaurin’s expansion.
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Bridge of Don Academy – Department of Mathematics
4.
Express
f(x) = tan-1 x
5.
Express
f(x) = (1 + x)4
Advanced Higher: Unit 3 – Sequences and Series
as a Maclaurin’s expansion.
as the first five terms of a Maclaurin’s expansion.
[Note that f(5)(0) is undefined so we can only do the first five terms.]
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Bridge of Don Academy – Department of Mathematics
f(x) = ln (1 + x)
Advanced Higher: Unit 3 – Sequences and Series
as a Maclaurin’s expansion.
6.
Express
7.
Find the first six terms in the Maclaurin’s expansion of f ( x)  (1  x) 2 .
1
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Bridge of Don Academy – Department of Mathematics
Advanced Higher: Unit 3 – Sequences and Series
More Examples:
8.
Expand f ( x)  e3x in ascending powers of x as far as the x 4 .
9.
Expand f ( x)  ln(1  2 x) in ascending powers of x as far as the x 5 .
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Bridge of Don Academy – Department of Mathematics
Advanced Higher: Unit 3 – Sequences and Series
10.
Expand f ( x)  sin 3x in ascending powers of x as far as the x 5 .
11.
Expand f ( x)  tan 2 x in ascending powers of x as far as the x 5 .
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Bridge of Don Academy – Department of Mathematics
12.
Advanced Higher: Unit 3 – Sequences and Series
Use the Maclaurin’s series to find the first five terms of an expansion for ln (2 + x).
Maclaurin’s Theorem is better remembered as follows,
1
2
3
f ( x)  f (0)  x f (0)  x f (0)  x f (0)  .....
1!
2!
3!



r 0
x r f ( r ) (0)
r!
Now try:
Page 91, Exercise 4 – question 6
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Bridge of Don Academy – Department of Mathematics
Advanced Higher: Unit 3 – Sequences and Series
CONVERGENCE (in brief)
If
Sn  S
as
n
then the series is said to converge.
Roughly speaking, we need the terms to become smaller as n increases. The easiest way to ensure
this is –1 < x < 1 (compare this with a geometric series).
However, a factorial in the denominator will eventually compensate for the power of x on top and
will give a convergent series for all x.
Here is a list of the more common series and their domains of validity for convergence:
(i )
2
3
4
e x  1  x  x  x  x  .....
2! 3! 4!
x
(ii )
3
5
7
sin x  x  x  x  x  .....
3! 5! 7!
x
(iii )
2
4
6
cos x  1  x  x  x  .....
2! 4! 6!
x
(iv)
2
3
4
ln(1  x)  x  x  x  x  .....
2
3
4
1  x  1
(v )
3
5
7
tan 1 x  x  x  x  x  .....
3
5
7
1  x  1
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Bridge of Don Academy – Department of Mathematics
Advanced Higher: Unit 3 – Sequences and Series
COMPOSITE FUNCTIONS
Examples:
13.
Express
sin 2x as a power series in x.
3
5
7
sin x  x  x  x  x  .....
3! 5! 7!

14.
see (ii) above.
(2 x)3 (2 x)5 (2 x) 7


 .....
3!
5!
7!
3
5
7
 2 x  8 x  32 x  128 x  .....
6
120
5040
3
5
7
 2 x  4 x  4 x  8 x  .....
3
15
315
sin(2 x)  (2 x) 
Express esin x as a power series in x (as far as the term in x4).
3
5
7
sin x  x  x  x  x  .....
3! 5! 7!
2
3
4
e x  1  x  x  x  x  .....
2! 3! 4!

see (ii) above
see (i) above
esin x =
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Bridge of Don Academy – Department of Mathematics
15.
Expand e x sin x as far as the term x 5 .
16.
Expand ln 1  e x as far as the term x 4 .

Advanced Higher: Unit 3 – Sequences and Series

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Bridge of Don Academy – Department of Mathematics
Advanced Higher: Unit 3 – Sequences and Series
ITERATION
Let us consider the roots of y = x2 – 4x – 8 by constructing a table of values.
x
-5
-4
-3
-2
-1
0
1
2
3
y
37
24
13
4
-3
-8
-11
-12
-11
We can see that there is a root between x = -2 and x = -1 (i.e. the curve must cross the x-axis).
Take the rearrangement of
x2  4x  8  0
as
x  1 x2  2
4
expressed as a recurrence relation
as xn 1  1 xn 2  2 .
4
From the table we know that there is a root between x = -2 and x = -1 so:
x0  1
x1  1 (1) 2  2  1 3
4
4
x2  1 (1 3 ) 2  2  1 15
4
4
64
etc. converges to x  1 464
This “search” can be illustrated graphically:
y=x
y  1 x2  2
4
As we can see, we spiral into the root.
We call this pattern a cobweb.
x0
Note that x2 – 4x – 8 = 0 could have been rearranged in different ways:
x
8
or x  4 x  8 .
x4
To calculate all of the roots of y = x2 – 4x – 8 we would need to use more than one formula i.e. no
one formula gives all the roots of y = x2 – 4x – 8.
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Bridge of Don Academy – Department of Mathematics
Advanced Higher: Unit 3 – Sequences and Series
An extension of the table of values tells us that there is another root between x = 5 and x = 6.
When trying to find this root we would need to try another arrangement / recurrence relation.
Let us use x  4 x  8 i.e. the recurrence relation xn1  4xn  8
and we would get a different graph.
We call this pattern a staircase.
y=x
y  4x  8
x0
A sequence given by a recurrence relation xn1  g( xn ) will converge to a root, , if and only if
g ( )  1 .
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Bridge of Don Academy – Department of Mathematics
Advanced Higher: Unit 3 – Sequences and Series
Example:
17.
Solve the equation x – sin(x + 1) = 0 correct to three decimal places using the iterative
sequence defined by xn1  sin( xn  1) where 0  x   .
Will this sequence converge?
g ( x)  sin( x  1)
 g ( x)  cos( x  1)
 g ( x ) will be 1 iff x    1
Provided x    1 isn’t the root then the sequence will converge.
Put x    1 into x – sin(x + 1) = (  1)  sin(  1  1)    1  0    1
 x    1 is not a root and so this rearrangement will work.
Let
x0  0
x1  sin(1  1)  0.90930
x2  sin(0.90930  1)  0.94325
x3  0.93144
x4  0.93567
x5  0.93417
x6  0.93470
x7  0.93451
x8  0.93458
x9  0.93456
We can see that, to three decimal places, the sequence has converged to x = 0.935.
Check that it has converged g ( x)  cos(0.935  1)  0.356
 g ( x)  1
Let us check, substitute x = 0.935 into the original equation, x – sin(x + 1) = 0, to give
0.935 – sin(0.935 + 1) = 0.000592
which is equal to zero when rounded to three decimal places and so this solution is valid.
There is a root to the equation x – sin(x + 1) = 0 at x = 0.935.
Now try:
Page 99, Exercise 8 – questions 1a, 2a, 2b, 5a
Page 102, Exercise 9 – question 3a – ignore mention of “1st order process”
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Bridge of Don Academy – Department of Mathematics
Advanced Higher: Unit 3 – Sequences and Series
Further examples (from SQA exams):
18.
Find the first four terms in the Maclaurin series for (2  x) ln(2  x) .
4
19.
Find the Maclaurin expansion of


f ( x)  ln  cos x  , 0  x   ,
2
as far as the term x 4 .
5
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Bridge of Don Academy – Department of Mathematics
20.
Advanced Higher: Unit 3 – Sequences and Series
A recurrence relation is defined by the formula


xn 1  1 xn  7 .
2
xn
Find the fixed points of this recurrence relation.
3
21.
Obtain the Maclaurin series for f ( x)  sin 2 x up to the term x 4 .
4
2
4
Hence write down a series for cos x up to the term x .
1
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Bridge of Don Academy – Department of Mathematics
22.
Advanced Higher: Unit 3 – Sequences and Series
Obtain the first three non-zero terms in the Maclaurin expsnasion of f ( x)  e x sin x .
5
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Bridge of Don Academy – Department of Mathematics
Advanced Higher: Unit 3 – Sequences and Series
JEAN-LE-ROND D'ALEMBERT (1717-1783)
Jean-le-Rond d'Alembert (1717-1783) worked so diligently in an
effort to prove the fundamental theorem of algebra (that every
polynomial equation f(x) = 0 having complex coefficients and of
degree n has at least one complex root) that the theorem is today
known in France as d'Alembert's theorem.
A famous and oft-quoted remark made by D'Alembert (and well
worth citing on occasion in an elementary algebra class) is:
"Algebra is generous; she often gives more than is asked of
her."
He also once aptly remarked: "Geometrical truths are in a way
asymptotic to physical truths; that is to say, the latter approach
the former indefinitely near without ever reaching them exactly."
Perhaps the most perceptive of D'Alembert's comments on
mathematics is the following: "I have no doubt that if men
lived separate from each other, and could in such a situation
occupy themselves about anything but self-preservation,
they would prefer the study of the exact sciences to the
cultivation of the agreeable arts. It is chiefly on account of
others that a man aims at excellence in the latter; it is on
his own account that he devotes himself to the former. In a
desert island, accordingly, I should think that a poet could
scarcely be vain, whereas a mathematician might still enjoy
the pride of discovery."
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