A: \({1 \over {1 + {x^2}}} = \sum\limits_{n = 0}^\infty { { {( - 1)}^n}{x^{2n}}} \matrix{ {} & {} \cr } ( - \infty < x < + \infty )\)
B: \({1 \over {1 + {x^2}}} = \sum\limits_{n = 0}^\infty { { {( - 1)}^n}{x^{2n}}} \matrix{ {} & {} \cr } ( - 1< x < 1)\)
C: \({1 \over {1 + {x^2}}} = \sum\limits_{n = 0}^\infty { { {( - 1)}^n}{x^{2n}}} \matrix{ {} & {} \cr } ( - 1 < x < 1)\)
D: \({1 \over {1 + {x^2}}} = \sum\limits_{n = 0}^\infty { { x^{2n}}} \matrix{ {} & {} \cr } ( - 1 < x < 1)\)
举一反三
- 将\(f(x)=e^x\)展开成\((x-3)\)的幂级数为( )。 A: \(\sum\limits_{n = 0}^\infty { { { { {(x - 3)}^n}} \over {n!}}} \matrix{ {} & {} \cr } ( - 1, 1)\) B: \({e^3}\sum\limits_{n = 0}^\infty { { { { {(x - 3)}^n}} \over {n!}}} \matrix{ {} & {} \cr } ( - 1, 1)\) C: \(\sum\limits_{n = 0}^\infty { { { { {(x - 3)}^n}} \over {n!}}} \matrix{ {} & {} \cr } ( - \infty , + \infty )\) D: \({e^3}\sum\limits_{n = 0}^\infty { { { { {(x - 3)}^n}} \over {n!}}} \matrix{ {} & {} \cr } ( - \infty , + \infty )\)
- 将函数\(f(x) = {e^x}\)展开成\(x\)的幂级数为（ ）。 A: \({e^x} = \sum\limits_{n = 0}^\infty { { { { x^n}} \over {n!}}} ( - \infty < x < + \infty )\) B: \({e^x} = \sum\limits_{n = 0}^\infty { { {( - 1)}^n} { { {x^n}} \over {n!}}} ( - \infty < x < + \infty )\) C: \({e^x} = \sum\limits_{n = 0}^\infty { { { { x^n}} \over {n!}}} ( - 1 < x < 1)\) D: \({e^x} = \sum\limits_{n = 0}^\infty { { {( - 1)}^n} { { {x^n}} \over {n!}}} ( - 1 < x < 1)\)
- 将\(f(x) = {1 \over {2 - x}}\)展开成\(x \)的幂级数为（ ）。 A: \({1 \over {2 - x}} = \sum\limits_{n = 0}^\infty { { { { x^n}} \over { { 2^{n }}}}} \)，\(( - 2,2)\) B: \({1 \over {2 - x}} = \sum\limits_{n = 0}^\infty { { { { x^n}} \over { { 2^{n }}}}} \)，\(\left( { - 2,2} \right]\) C: \({1 \over {2 - x}} = \sum\limits_{n = 0}^\infty { { { { x^n}} \over { { 2^{n + 1}}}}} \)，\(( - 2,2)\) D: \({1 \over {2 - x}} = \sum\limits_{n = 0}^\infty { { { { x^n}} \over { { 2^{n + 1}}}}} \)，\(\left( { - 2,2} \right]\)
- 设幂级数\(\sum\limits_{n = 0}^\infty { { a_n}} {x^n}\)与\(\sum\limits_{n = 1}^\infty { { b_n}{x^n}} \)的收敛半径分别为\( { { \sqrt 5 } \over 3}\)与\({1 \over 3}\)，则幂级数\(\sum\limits_{n = 1}^\infty { { {a_n^2} \over {b_n^2}}} {x^n}\)的收敛半径为( )。 A: 5 B: \( { { \sqrt 5 } \over 3}\) C: \({1 \over 3}\) D: \({1 \over 5}\)
- 下列级数中,收敛的是（ ）. A: \(<br/>\sum\limits_{n = 1}^\infty { { 1 \over n}} \) B: \(<br/>\sum\limits_{n = 1}^\infty { { 1 \over { { n^2}}}} \) C: \(<br/>\sum\limits_{n = 1}^\infty { { 1 \over {\sqrt n }}} \) D: \( \sum\limits_{n = 1}^\infty { { 1 \over {\root 3 \of { { n^2}} }}} \)
内容
- 0
下面级数求和错误的是 A: $\sum_{n=0}^\infty q^n = \frac{1}{1-q} (0\lt q\lt1) $ B: $\sum_{n=1}^\infty \frac{x^{2^{n-1}}}{1-x^{2^n}} = \frac{x}{1-x} (|x|\lt 1) $ C: $\sum_{n=1}^\infty \frac{1}{{n!}} = e $ D: $\sum_{n=1}^\infty \frac{x^{2^{n-1}}}{1-x^{2^n}} = \frac{1}{1-x} (x>1) $
- 1
在其定义区间上连续的函数是( )。 A: \(f(x) = \left\{ {\matrix{ {x\quad ,{\rm{0}} \le x \le {\rm{1}}} \cr {1 - x\quad ,1 < x \le 2} \cr } } \right.\) B: \(f(x) = \left\{ {\matrix{ {x\quad ,0 < x \le 1 } \cr {2 - x\quad ,1 < x \le 2} \cr } } \right.\) C: \(f(x) = \left\{ {\matrix{ {x\;\quad ,0 \le x < 1} \cr {0\;\quad \quad ,x = 1} \cr {2 - x\quad ,1 < x \le 2} \cr } } \right.\) D: \(f(x) = \left\{ {\matrix{ { { 1 \over {x - 1}}\quad ,0 \le x \le 1} \cr {0\quad ,1 \le x \le 2} \cr } } \right.\)
- 2
函数\(f(x) = \left\{ {\matrix{ { { x^2} - 1\;, - 1 \le x < 0} \cr {x\;\quad \;,0 \le x < 1} \cr {2 - x\;\quad ,1 \le x \le 2} \cr } } \right.\)在\(x =\)（ ）处间断。______
- 3
\( {1 \over {1 + x}} \)的麦克劳林公式为（ ）。 A: \( {1 \over {1 + x}} = 1 + x + { { {x^2}} \over 2} + \cdots + { { {x^n}} \over {n!}} + o\left( { { x^n}} \right) \) B: \( {1 \over {1 + x}} = 1 + x + {x^2} + \cdots + {x^n} + o\left( { { x^n}} \right) \) C: \( {1 \over {1 + x}} = 1 - x + {x^2} - \cdots + {( - 1)^n}{x^n} + o\left( { { x^n}} \right) \) D: \( {1 \over {1 + x}} = 1 - x - { { {x^2}} \over 2}- \cdots - { { {x^n}} \over {n!}} + o\left( { { x^n}} \right) \)
- 4
\( \sin x \)的麦克劳林公式为（ ）. A: \( \sin x = x - { { {x^3}} \over {3!}} + { { {x^5}} \over {5!}} - \cdots + {( - 1)^n} { { {x^{2n + 1}}} \over {\left( {2n + 1} \right)!}} + o\left( { { x^{2n + 2}}} \right) \) B: \( \sin x = 1 - { { {x^2}} \over {2!}} + { { {x^4}} \over {4!}} - { { {x^6}} \over {6!}} + \cdots + {( - 1)^n} { { {x^{2n}}} \over {\left( {2n} \right)!}} + o\left( { { x^{2n + 1}}} \right) \) C: \( \sin x = 1 + x + { { {x^2}} \over 2} + \cdots + { { {x^n}} \over {n!}} + o\left( { { x^n}} \right) \)