7.3 Binomial Theorem with Rational Power 指数为有理数的二项式定理 - 独中课程 | 线上补习

高二数学 | 高级数学

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7.3 Binomial Theorem with Rational Power 指数为有理数的二项式定理

[1]

Expand the following in ascending powers of x, as far as the terms in $x^3$ ; and state the ranges of values of x for which the expansions are valid.

按x的升幂展开以下各式至 $x^3$ 项, 并写x的限制范围

(a) (1 + x) $^{-2}$

(b) $\sqrt{2-x}$

(d) (8 – 3x) $^{-1/2}$

Answer:

(a) 1 – 2x + 3x $^2$ – 4x $^3$ + …, $-1 < x < 1$

(b) $\sqrt2$ (1 – $\frac{1}{4}x$ – $\frac{1}{32}x^2$ – $\frac{1}{128}x^3$ – $\ldots$ ), $-2 < x < 2$

(d) $\frac{1}{2\sqrt2}$ (1 + $\frac{3}{16}x$ + $\frac{27}{512}x^2$ + $\frac{135}{8192}x^3$ + $\ldots$ ) , $-\frac{8}{3}$ < x < $\frac{8}{3}$

[2]

Calculate the coefficient of $x^{-\frac{9}{2}}$ of the expansions of $(x-\frac{2}{3}x^{-\frac{1}{3}})^{-\frac{1}{2}}$ .

求 $(x-\frac{2}{3}x^{-\frac{1}{3}})^{-\frac{1}{2}}$ 在的展开式中 $x^{-\frac{9}{2}}$ 的系数.

Answer:

$\frac{5}{54}$

[3]
Calculate the coefficient of $x^{10}$ of the expansions of $(1 - 2x)^{-3}$ in ascending powers of x.

求 $x^{10}$ 在求 $(1 - 2x)^{-3}$ 的升幂展开式中的系数.的升幂展开式中的系数.

Answer:

67584

[4]

Prove that the coefficient of $x^r$ in the expansion of (1 – 4x) $^{-\frac{1}{2}}$ is $\frac{(2r)!}{(r!)^2}$ .

试证 $x^r$ 在(1 – 4x) $^{-\frac{1}{2}}$ 的展开式中的系数是 $\frac{(2r)!}{(r!)^2}$ .

[5]

Show that the n $^{th}$ coefficient in the expansion of (1 – x) $^{-n}$ is double of the (n – 1) $^{th}$ .

试证 (1 – x)-n的展开式中第n项的系数是第n – 1项的两倍.

[6]

Calculate the coefficient of $x^{-\frac{5}{3}}$ in the expansion of $(x-\frac{1}{3}x^\frac{1}{2})^\frac{1}{3}$ .

求在 $(x-\frac{1}{3}x^\frac{1}{2})^\frac{1}{3}$ 的展开式中 $x^{-\frac{5}{3}}$ 的系数。

Answer:

$-\frac{10}{19683}$

[7]

Find the first four terms of the expansions of $\frac{x+2}{(1+x)^3}$ in ascending powers of x.

按x的升幂展开 $\frac{x+2}{(1+x)^3}$ 的首四项

Answer:

2 – 5x + 9x $^2$ – 14x $^3$ + …

[8]

Express f(x)= $\frac{1+x}{(1-x)(2+x)}$ in partial fraction. Hence, or otherwise, express f (x) as a series of ascending powers of x up to and including the term in x $^3$ . State the range of values of x for which the expansion is valid.

将 f(x)= $\frac{1+x}{(1-x)(2+x)}$ 化为部分分式. 据此或用其他方法, 依x的升幂展开f (x)至x $^3$ 项. 写出x值的范围使此展开式有意义

Answer:

$\frac{2}{3(1-x)}$ – $\frac{1}{3(2+x)}$ ,

$\frac{1}{2}$ + $\frac{3}{4}x$ + $\frac{5}{8}x^2$ + $\frac{11}{16}x^3$ + $\ldots$ ,

$-1 < x < 1$