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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