CHAPTER
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14: THE BINOMIAL EXPANSION
Miscellaneous exercise 14
195
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1 Find the series expansion of (1+2x)~ up to and including the term in .x 3 , simplifying the (OCR) coefficients.
2 Expand (1- 4x)! as a series of ascending powers of x, where Ix I<!, up to and including the term in x 3 , expressing the coefficients in their simplest form. (OCR)
!,
3 Expand (1+2x )-3 as a series of ascending powers of x, where I x I < up to and including the term in x3, expressing the coefficients in their si°mplest form.
4 Expand
1
(1+2x 2)
?.
(OCR)
as a.series in ascending powers of x, up to and including the term in
x 6 , giving the coefficients in their simplest form.
(OCR)
.
I
S Obtain the first three terms in the expansion, in ascending powers of x, of (4 + x )2 . State the set of values of x for which the expansion is valid. (OCR) 3
6 If x is small compared with a , expand a 3 in ascending powers of ~ up to and a 4 (a2+x2)2 including the term in x 4 . (OCR) a 7 Given that Ix I< 1, expand .,)1 + x as a series of ascending powers of x, up to and including the term irt x 2 . Show that, if x is small, then (2 - x ).,JI+ x ,,,·a+ bx 2 , where the values of a a~d b are to be stated. . (OCR) 8 Expand (1- xf .as a series of ascending powers of x, given that Ix I< 1. Hence express 2
1
+x
0-zj 2
in the form 1+3x + ax 2 + bx 3 + ... , where the values of a and b are to be stated.
.
(OCR)
\
. \\
9 Obtain the first three terms in the expansion, in ascending powers of\ of (8 + 3x )% , stating the set of values of x for which the expansion is valid.
\
(OCR)
10 Write down the first four terms of the series expansion in ascending powers of x of (1 :-- x)!, simplifying the coefficients. By taking x = 0.1, use your answer to show that
V900 "" 1;6~~ 1 .
(OCR)
11 Give the binomial expansion, for small x, of (1 + x)± up to and including the term in x 2 , and simplify the coefficients. By putting x = in your expression, show that
f6
~,,,~17 4096.
(Offi) 6
2+(1+lx) 12 Expand + ~ .in ascending powers of x up to and including the term in x 2 . 2
3
(OCR)