# SERIES EXPANSIONS

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```					                                           SERIES EXPANSIONS

1)    Expand the following in ascending powers of x.
a) (1 + x)4,         b) (1 x)3,            c) (1 2x)3,                            d) (2 + x)4,
e) (1 + 2 x ) 3 ,
1
f) (2 x)4,            g) (2  3x ) 3 .

2)    Expand (2a + x)4 in ascending powers of x.

3)    Write down the term in x3 in the expansions of : a) (1 + x)10,                    b) (1 x ) 5 ,    c) (2 + 3x)7.

4)    Expand the following in ascending powers of x up to and including the third term.
a) (1 + 2x)10,       b) (1    1   8
2 x) ,       c) (3 2 x ) 12 .

5)    Expand (1 + x)3 as a series in x.
Hence expand (1 + 2x ) × (1 + x ) 3 .

6)              (
i) Expand 1 +      )3   3 , leaving surds in your answer.
ii) Expand (1     3 ) , leaving surds in your answer.
3

iii) Use your answers from i) and ii) to simplify (1 +        3   ) 3 + (1            )3
3 .

4
2
7)    Find the term independent of x in the expansion of x 2 +                      .
x2

8)    Expand the following as far as the term in x3. In each case give the range of values of x for which the
expansion is valid.
1                        1
a) (1 + x ) 3 ,             b) (1   x) 2 ,        c) (1 + 2 x ) 2 ,                        d) (1       2 x) 2 ,
1
e) 1 +     1
2   x,           f)              ,     g) x 1 + 2 x ,                           h) (1       1
x) 2 .
(1 + 4 x ) 3                                                            3

9)    Expand      1 + x 2 in ascending powers of x up to and including the term in x6.
1

{Hint put y = x2 and expand (1 + y ) 2 etc.}

(1 + x )
10)   Expand                  in ascending powers of x up to and including the term in x4.
1    x

11)   Expand (1 + ax ) 2 in ascending powers of x up to and including the term in x3.
Hence write down the values of a, P and Q such that the first four terms in the expansion of
(1 + ax ) 2 are 1 + Px + Qx 2      1 3
2 x .

12)   Expand (1    x ) 2 in ascending powers of x up to and including the term in x3, given that x < 1.
(3 + x)                           2    3
Hence express           2 in the form 3 + 7 x + ax + bx + …… where the values of a and b are to be
(1 x )
stated.

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13)    Find the first three non-zero terms in the expansion of            4 + y and write down the range of values
y
of y for which the expansion is valid.                   {Hint   4 + y = 4× 1 +          4   etc.}

1

14)    Obtain the expansion of (16 + y ) 2 in ascending powers of y up to and including the term in y2.
k   3k 2
Hence show that 16 + 4k + k 2               4 +        +      .               {Hint put y = 4k + k2.}
2    32

1
*15)   Expand    1 + x and           in ascending powers of x as far as the terms in x3.
1 x
1 + x
Hence expand        as far as the term in x3.
1 x

x2      x4     x6
16)    Verify the following Maclaurin expansions. a) cos x = 1         +             + ......
2!      4!     6!
x2      x3     x4
b) e x = 1 + x +       +      +       + ......
2!      3!     4!

{Miscellaneous questions.}
dy d 2 y       d3y
17)    Given that y = sin( + x ) , find    ,     and        .
d x dx 2       dx 3
Hence find the Maclaurin series for sin( + x ), up to and including the term in x3.

1
18)    Find the Maclaurin series for y =               , up to and including the term in x3.
1       x

19)    Find, from first principles, the Maclaurin expansion for y = ln(1 x ), up to and including the term
in x3.
Describe how this series could have been obtained from the series for ln (1 + x).

20)    Using the standard expansions for ex, sin x, cos x and ln (1 + x), find series expansions for the
following functions, up to and including the terms in x4.
x
a) e 3 x ,            b) cos 2x,              c) sin      ,            d) 2sin x cos x,
2
1 x                                             1
e) ln         ,       f) ln 1 + 2x {Hint ln( ) 2 = 1 ln( ) etc.}
2
1 + x

d2 y
21)    Given that y = cos       + 2x , find             .
3                  dx 2
Hence obtain the Maclaurin series for cos                 + 2x , up to and including the term in x2.
3

22)    Given that x < 1 , expand (1 + x ) as a series of ascending powers of x, up to and including the
term in x2. {Hint binomial theorem!}
Show that, if x is small, then (2 x ) (1 + x )     a + bx 2 , where the values of a and b are to be
stated.

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dy                 d2 y         9
23)   Given that y = ln (4 + 3x), find   and show that          =        when x = 0.
dx                 dx 2        16
Hence, or otherwise, obtain the Maclaurin series for ln (4 + 3x), up to and including the term in x2.

3                                                  ex                    x2
24)   If x is so small that x and higher powers of x may be neglected, show that                     1 +    .
(1 + x )                 2
ex
{Hint          = e x × (1 + x )          1
etc.}
(1 + x )

25)   Find the values of the constants A, B and C such that the series expansion of Acos x + Bex + C is the
same as the series expansion of 4 (1 + x ) , given that x is so small that terms in x3 and higher
powers of x may be neglected.

26)   Find the first 3 terms in the series expansion, in ascending powers of x, of
i) e ax ,       ii) (1 + 3x)b, for 1 < x < 1 .
3           3
1            1
Given that        3   < x <    3   and the first non-zero term in the series expansion, in ascending powers
ax                b
of x, of e            (1 + 3x ) is 6x2, find the values of a and b.

27)   If x is so small that x3 and higher powers of x may be neglected, show that
1     2x
ln            = ex       e 2x .
1 x

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ANSWERS. {Where appropriate, answers given are simplified as far as possible.}
1) a) 1 + 4x + 6x2 + 4x3 + x4, b) 1 3x + 3x 2 x 3 , c) 1   6 x + 12 x 2    8x 3 ,
3             3           1
d) 16 + 32x + 24x2 + 8x3 + x4, e) 1 +                                      2   x +       4   x2 +    8   x 3 , f) 16            32 x + 24 x 2            8x 3 + x 4 ,
g) 8 36 x + 54 x 2       27 x 3 .
4       3       2 2
2) 16a + 32a x + 24a x + 8ax3 + x4.
3) a) 120x3, b) 10x3, c) 15120x3.
4) a) 1 + 20x + 180x2, b) 1      4 x + 7 x 2 , c) 531441     4251528 x + 15588936 x 2 .
5) (1 + x)3 = 1 + 3x + 3x2 + x3.       (1 + 2x ) × (1 + x ) 3 = 1 + 5x + 9x2 + 7x3 + 2x4.
6) i) 10 + 3 3 +             ( 3) 3 ,         ii) 10              3 3              ( 3) 3 ,      iii) 20.
7) 24.
8) a) 1 + 1 x3
1
9 x
2
+ 81 x 3
5
...... ; x < 1, i. e. 1 < x < 1 .
b) 1      1
2 x
1 2
8 x
1
16 x
3
...... ; x < 1, i. e. 1 < x < 1 .
c) 1    4 x + 12 x   2
32 x + ...... ; x < 1 , i. e. 1 < x < 1 .
3
2        2        2

d) 1 + 4 x + 12 x + 32 x + ...... ; x < 2 , i. e. 2 < x < 2 .
2             3                    1         1        1

e) 1 + 1 x4
1
32 x
2
+ 128 x 3
1
...... ; x < 2, i. e. 2 < x < 2 .
f) 1 12 x + 96 x      2
640 x + ...... ; x < 1 , i. e. 1 < x < 1 .
3
4        4        4

g) x + x    2  1
2 x
3
+ ...... ; x < 2 , i.e. 2 < x < 2 .
1           1         1

h) 1 + 3 x + 3 x + 27 x 3 + ...... ; x < 3, i. e. 3 < x < 3 .
2      1 2           4

9) 1 + 2 x 2
1       1
8 x
4
+ 16 x 6
1
......
10) 1 + 1 5x + 0 875x + 0 6875x 3 + 0 5859x 4 + ......
2

11) 1 2 ax + 3a 2 x 2             4a 3 x 3 + ...... a = 1 , P = 1, Q = 4 .
2
3

12) 1 + 2x + 3x2 + 4x3 + ……; a = 11, b = 15.
13) 2 + 4 y1      1
64 y
2
+ ......; y < 4, i.e. 4 < y < 4 .
14) 4 +      1
8   y      1
512   y 2 + ......
x2               x3                               1
15) 1 + x = 1 +                  x
2        8        +       16               ......,                 = 1 + x + x2 + x3 + …….
1       x
1 + x                              11 x 2                23s 3
= 1 +            3x
2      +      8          +         16         + ......
1 x
x3
17) sin(     + x) =              x +                           ......
3!
1
18)      = 1 + x + x2 + x3 + ……
1 x
1 2    1 3
19) ln(1 x ) =    x     2 x    3x                                             ......
9       2        9       3             27       4                                                                         1            1
20) a) 1 + 3x +          2   x       +    2   x        +         8   x        + ......, b) 1                2x 2 +    2
3   x4         ...... , c)   2   x        48   x 3 + ...... ,
4
d) 2 x         3   x 3 + ......, e)                  2x               2
3   x 3 + ...... , f) x               x2 +        4
3   x3         2 x 4 + ......
d2 y                                                                                              1
21)     2
=        4 cos              + 2x .                      cos           + 2x             =            3x           x 2 + ......
dx             3                                                           3                         2
22) (1 + x ) = 1 + 1 x
2
1
8   x   2
+ ......               (2        x ) (1 + x )             2         3
4   x 2 . a = 2, b =         3
4.
3        9 2
23) ln (4 + 3x) = ln 4 + 4 x      32 x + ......
25) A = 3, B = 2, C = 1.
9                                                         4
26) i) 1 + ax + 2 a 2 x 2 + ...... . ii) 1 + 3bx +
1
2 b (b       1) x 2 + ......            a = 4, b =        3.

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