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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. www.mathsguru.co.uk 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. www.mathsguru.co.uk 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 www.mathsguru.co.uk 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. www.mathsguru.co.uk

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Series Expansions, series expansion, Power Series, taylor series, Maclaurin series, radius of convergence, n + 1, vector ﬁeld, Taylor Series Expansions, power series expansion

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posted: | 2/4/2011 |

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