若函数$f(x)$满足条件：$f(x+\pi)=-f(x)$, 则在$(-\pi,\pi)$内的傅里叶级数满足下列哪个特性？
A: $a_{2n}=b_{2n}=0, (n=1,2,\cdots)$
B: $a_{2n-1}=b_{2n-1}=0, (n=1,2,\cdots)$
C: $a_{2n-1}=b_{2n}=0, (n=1,2,\cdots)$
D: $a_{2n}=b_{2n-1}=0, (n=1,2,\cdots)$
A: $a_{2n}=b_{2n}=0, (n=1,2,\cdots)$
B: $a_{2n-1}=b_{2n-1}=0, (n=1,2,\cdots)$
C: $a_{2n-1}=b_{2n}=0, (n=1,2,\cdots)$
D: $a_{2n}=b_{2n-1}=0, (n=1,2,\cdots)$
举一反三
- 计算\({\oint_L {({x^2} + {y^2})} ^n}ds\)，其中\(L\)为圆周\(x = a\cos t\)，\(y=asint\)\((0 \le t \le 2\pi )\)。 A: \(2\pi {a^{n + 1}}\) B: \(2\pi {a^{2n + 1}}\) C: \(\pi {a^{n + 1}}\) D: \(2\pi {a^{n + 1}}\)
- \( \sin x \)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: \( \sin x = 1 + x + { { {x^2}} \over 2} + \cdots + { { {x^n}} \over {n!}} + o\left( { { x^n}} \right) \)
- 下列多项式在有理数域上不可约的是（ ）。 A: $(x-a_{1})(x-a_{2})...(x-a_{n})-1$，其中$a_{1},a_{2},...,a_{n}$是两两互异的整数； B: $(x-a_{1})(x-a_{2})...(x-a_{n})+1$，其中$a_{1},a_{2},...,a_{n}$是两两互异的整数； C: $(x-a_{1})^{2}(x-a_{2})^{2}...(x-a_{n})^{2}+1$，其中$a_{1},a_{2},...,a_{n}$是两两互异的整数； D: $(x-a_{1})^{2}(x-a_{2})^{2}...(x-a_{n})^{2}-1$，其中$a_{1},a_{2},...,a_{n}$是两两互异的整数.
- \( {1 \over {1 + x}} \)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: \( {1 \over {1 + x}} = 1 - x - { { {x^2}} \over 2}- \cdots - { { {x^n}} \over {n!}} + o\left( { { x^n}} \right) \)
- 函数\(f(x) = x^2,\; x \in [-\pi,\pi]\)的Fourier级数为 A: \(\frac{\pi^2}{3}+4\Sigma_{n=1}^{\infty} \frac{(-1)^n}{n^2} \sin nx ,\; x \in [-\pi,\pi]\) B: \(\frac{\pi^2}{3}+4\Sigma_{n=1}^{\infty} \frac{(-1)^n}{n^2} \cos nx ,\; x \in [-\pi,\pi]\) C: \(\frac{2\pi^2}{3}+4\Sigma_{n=1}^{\infty} \frac{(-1)^n}{n^2} \sin nx ,\; x \in [-\pi,\pi]\) D: \(\frac{2\pi^2}{3}+4\Sigma_{n=1}^{\infty} \frac{(-1)^n}{n^2} \cos nx ,\; x \in [-\pi,\pi]\)