写在前面:
- 之前大唐杯的笔记有很多催更,最近长春疫情严重,学校封寝之后实在是难以保持高效的学习效率,这里说声抱歉。也希望同学们自己尝试去总结知识点,一个名词一个名词去百度,才有深刻的记忆~
- 本文仅用于为自己梳理知识点用,写的比较随心所欲,有一些过于简单的公式,一些过于偏,我在做题中从未做过的公式就不记录了。
- 本文是基于张宇基础30讲梳理的,基本按照其顺序排列,复习到了高数下之后明显感觉需要背的东西变多了,这里也是以高数下为重点。
- 数学一选手,后面还有2讲内容张宇没更新,以后再补。
- 写了一半字数超了,只能分两节发布了XD。
一、高等数学预备知识
1.函数奇偶性
(1) F ( x ) = f ( x ) − f ( − x ) F(x)=f(x)-f(-x) F(x)=f(x)−f(−x)必为奇函数, F ( x ) = f ( x ) + f ( − x ) F(x)=f(x)+f(-x) F(x)=f(x)+f(−x)必为偶函数.
(2)导数与积分奇偶性
∫ a x f ( t ) d t \int_{a}^{x} f(t)dt ∫axf(t)dt | f ( x ) f(x) f(x) | f ′ ( x ) {f}' (x) f′(x) |
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偶函数 | 奇函数 | 偶函数 |
奇函数 | 偶函数 | 奇函数 |
当 ∫ 0 T f ( x ) d x = 0 \int_{0}^{T} f(x)dx=0 ∫0Tf(x)dx=0时,周期为T | 周期为T | 周期为T |
(3)奇偶函数运算
奇 函 数 × 奇 函 数 = 偶 函 数 奇函数\times 奇函数 = 偶函数 奇函数×奇函数=偶函数
奇 函 数 × 偶 函 数 = 奇 函 数 奇函数\times 偶函数 = 奇函数 奇函数×偶函数=奇函数
偶 函 数 × 偶 函 数 = 偶 函 数 偶函数\times 偶函数 = 偶函数 偶函数×偶函数=偶函数
奇 函 数 + 奇 函 数 = 奇 函 数 奇函数 + 奇函数 = 奇函数 奇函数+奇函数=奇函数
偶 函 数 + 偶 函 数 = 偶 函 数 偶函数 + 偶函数 = 偶函数 偶函数+偶函数=偶函数
2.三角函数相关公式
a r c t a n α + a r c c o t α = π 2 arctan\alpha+arccot\alpha =\frac{\pi}{2} arctanα+arccotα=2π
1 + t a n 2 α = s e c 2 α 1+tan^{2}\alpha =sec^{2}\alpha 1+tan2α=sec2α
1 + c o t 2 α = c s c 2 α 1+cot^{2}\alpha =csc^{2}\alpha 1+cot2α=csc2α
s i n 3 α = − 4 s i n 3 α + 3 s i n α sin3\alpha=-4sin^{3} \alpha+3sin\alpha sin3α=−4sin3α+3sinα
c o s 3 α = 4 c o s 3 α − 3 c o s α cos3\alpha=4cos^{3} \alpha-3cos\alpha cos3α=4cos3α−3cosα
s i n ( α ± β ) = s i n α c o s β ± c o s α s i n β sin(\alpha\pm\beta)=sin\alpha cos\beta\pm cos\alpha sin \beta sin(α±β)=sinαcosβ±cosαsinβ
c o s ( α ± β ) = c o s α c o s β ∓ s i n α s i n β cos(\alpha\pm\beta)=cos\alpha cos\beta \mp sin\alpha sin \beta cos(α±β)=cosαcosβ∓sinαsinβ
t a n ( α ± β ) = t a n α ± t a n β 1 ∓ t a n α t a n β tan(\alpha\pm\beta)=\frac{tan\alpha\pm tan\beta}{1\mp tan\alpha tan\beta } tan(α±β)=1∓tanαtanβtanα±tanβ
万能公式: u = t a n x 2 ⇒ s i n x = 2 u 1 + u 2 , c o s x = 1 − u 2 1 + u 2 u=tan\frac{x}{2} \Rightarrow sinx=\frac{2u}{1+u^{2} } ,cosx=\frac{1-u^{2}}{1+u^{2}} u=tan2x⇒sinx=1+u22u,cosx=1+u21−u2
3.数列、因式分解
(1)等比数列
等比数列 | a 1 , a 1 r , a 1 r 2 , ⋯ , a 1 r n − 1 a_{1} ,a_{1}r,a_{1}r^{2} ,\cdots ,a_{1}r^{n-1} a1,a1r,a1r2,⋯,a1rn−1 |
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通项 | a n = a 1 r n − 1 a_{n}=a_{1}r^{n-1} an=a1rn−1 |
前n项和 | S n = a 1 ( 1 − r n ) 1 − r S_{n}=\frac{a_{1}(1-r^{n} ) }{1-r} Sn=1−ra1(1−rn) ( r ≠ 1 ) (r≠1) (r=1) |
当 ∣ r ∣ < 1 \mid r\mid < 1 ∣r∣<1时 | 有 lim n → ∞ a n = 0 \lim_{n \to \infty} a_{n} =0 limn→∞an=0 |
S n = a 1 1 − r S_{n}=\frac{a_{1} }{1-r} Sn=1−ra1 | S n = 首 项 1 − 公 比 S_{n}=\frac{首项 }{1-公比} Sn=1−公比首项 |
(2)数列绝对值性质
lim n → ∞ a n = A ⇒ lim n → ∞ ∣ a n ∣ = ∣ A ∣ \lim_{n \to \infty} a_{n} =A\Rightarrow \lim_{n \to \infty} \mid a_{n}\mid =\mid A \mid limn→∞an=A⇒limn→∞∣an∣=∣A∣
lim n → ∞ a n = 0 ⇔ lim n → ∞ ∣ a n ∣ = 0 \lim_{n \to \infty} a_{n} =0\Leftrightarrow \lim_{n \to \infty} \mid a_{n}\mid =0 limn→∞an=0⇔limn→∞∣an∣=0
(3)3次幂的因式分解
( a + b ) 3 = a 3 + 3 a 2 b + 3 a b 2 + b 3 (a+b)^{3} =a^{3} +3a^{2} b +3ab^{2}+ b^{3} (a+b)3=a3+3a2b+3ab2+b3
( a − b ) 3 = a 3 − 3 a 2 b + 3 a b 2 − b 3 (a-b)^{3} =a^{3} -3a^{2} b +3ab^{2}- b^{3} (a−b)3=a3−3a2b+3ab2−b3
a 3 − b 3 = ( a − b ) ( a 2 + a b + b 2 ) a^{3}-b^{3} =(a-b)(a^{2} +ab+b^{2}) a3−b3=(a−b)(a2+ab+b2)
4.常用不等式
(1)基本不等式
a b ≤ a + b 2 ≤ a 2 + b 2 2 \sqrt{ab} \le \frac{a+b}{2} \le \sqrt{\frac{a^{2}+b^{2} }{2} } ab≤2a+b≤2a2+b2
∣ a b ∣ ≤ a 2 + b 2 2 \mid ab\mid \le \frac{a^{2}+b^{2} }{2} ∣ab∣≤2a2+b2
∣ ∫ a b f ( x ) d x ∣ ≤ ∫ a b ∣ f ( x ) ∣ d x \mid \int_{a}^{b} f(x) dx\mid \le \int_{a}^{b} \mid f(x)\mid dx ∣∫abf(x)dx∣≤∫ab∣f(x)∣dx
(2) x → 0 + x\to 0^{+} x→0+邻域的不等式
s i n x < x < t a n x sinx< x <tanx sinx<x<tanx
a r c t a n x < x < a r c s i n x arctanx< x <arcsinx arctanx<x<arcsinx
e x ≥ x + 1 e^{x} \ge x+1 ex≥x+1
ln x ≤ x − 1 \ln{x} \le x-1 lnx≤x−1
ln ( x + 1 ) ≤ x \ln{(x+1)} \le x ln(x+1)≤x
二、极限
1.常用泰勒公式 ( x → 0 ) (x\to0) (x→0)
s i n x = x − x 3 3 ! + o ( x 3 ) sinx=x-\frac{x^{3} }{3!} +o(x^{3}) sinx=x−3!x3+o(x3)
a r c s i n x = x + x 3 3 ! + o ( x 3 ) arcsinx=x+\frac{x^{3} }{3!} +o(x^{3}) arcsinx=x+3!x3+o(x3)
c o s x = 1 − x 2 2 ! + x 4 4 ! + o ( x 3 ) cosx=1-\frac{x^{2} }{2!}+\frac{x^{4} }{4!}+o(x^{3}) cosx=1−2!x2+4!x4+o(x3)
t a n x = x + x 3 3 + o ( x 3 ) tanx=x+\frac{x^{3} }{3} +o(x^{3}) tanx=x+3x3+o(x3)
a r c t a n x = x − x 3 3 + o ( x 3 ) arctanx=x-\frac{x^{3} }{3} +o(x^{3}) arctanx=x−3x3+o(x3)
l n ( 1 + x ) = x − x 2 2 + x 3 3 + o ( x 3 ) ln(1+x)=x-\frac{x^{2} }{2}+\frac{x^{3}}{3}+o(x^{3}) ln(1+x)=x−2x2+3x3+o(x3)
e x = 1 + x + x 2 2 ! + x 3 3 ! + o ( x 3 ) e^{x} =1+x+\frac{x^{2} }{2!}+\frac{x^{3}}{3!}+o(x^{3}) ex=1+x+2!x2+3!x3+o(x3)
a x = 1 + x l n a + l n 2 a 2 ! x 2 + l n 3 a 3 ! x 3 + o ( x 3 ) a^{x} =1+xlna+\frac{ln^2a}{2!}x^{2}+\frac{ln^3a}{3!}x^{3}+o(x^{3}) ax=1+xlna+2!ln2ax2+3!ln3ax3+o(x3)
( 1 + x ) α = 1 + α x + α ( α − 1 ) 2 ! x 2 + o ( x 2 ) (1+x)^{\alpha}=1+\alpha x+\frac{\alpha(\alpha - 1)}{2!} x^{2} +o(x^{2}) (1+x)α=1+αx+2!α(α−1)x2+o(x2)
2.无穷小定义
α ( x ) \alpha (x) α(x)是 β ( x ) \beta (x) β(x)的 | 定义 |
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高阶无穷小 | lim α ( x ) β ( x ) = 0 \lim \frac{\alpha (x)}{\beta (x)} =0 limβ(x)α(x)=0 |
低阶无穷小 | lim α ( x ) β ( x ) = ∞ \lim \frac{\alpha (x)}{\beta (x)} =\infty limβ(x)α(x)=∞ |
同阶无穷小 | lim α ( x ) β ( x ) = c ≠ 0 \lim \frac{\alpha (x)}{\beta (x)} =c\ne 0 limβ(x)α(x)=c=0 |
等价无穷小 | lim α ( x ) β ( x ) = 1 \lim \frac{\alpha (x)}{\beta (x)} =1 limβ(x)α(x)=1 |
k阶无穷小 | lim α ( x ) [ β ( x ) ] k = c ≠ 0 \lim \frac{\alpha (x)}{[\beta (x)]^k} =c\ne 0 lim[β(x)]kα(x)=c=0 |
3.常用等价无穷小 ( x → 0 ) (x\to0) (x→0)
l n ( 1 + x ) ∼ x ln(1+x)\sim x ln(1+x)∼x
e x − 1 ∼ x e^x-1\sim x ex−1∼x
a x − 1 ∼ x l n a a^x-1\sim xlna ax−1∼xlna
1 − c o s x ∼ 1 2 x 2 1-cosx \sim \frac{1}{2}x^2 1−cosx∼21x2
( 1 + x ) α − 1 ∼ α x (1+x)^\alpha-1 \sim \alpha x (1+x)α−1∼αx
x x + a ∼ x \frac{x}{x+a} \sim x x+ax∼x
4.常用极限运算
lim x → 0 + x α l n x = 0 \lim_{x \to 0^+} x^\alpha lnx=0 limx→0+xαlnx=0
lim x → ∞ x x + a = 1 \lim_{x \to \infty }\frac{x}{x+a}= 1 limx→∞x+ax=1
lim x → ∞ x α c x = 0 \lim_{x \to \infty }\frac{x^\alpha}{c^x} =0 limx→∞cxxα=0
5.几种左右极限不同的例子
极限 | 结果 |
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lim x → + ∞ e x \lim_{x \to +\infty} e^x limx→+∞ex | + ∞ +\infty +∞ |
lim x → − ∞ e x \lim_{x \to -\infty} e^x limx→−∞ex | 0 0 0 |
lim x → 0 + s i n x ∣ x ∣ \lim_{x \to 0^+}\frac{sinx}{\mid x \mid} limx→0+∣x∣sinx | 1 1 1 |
lim x → 0 − s i n x ∣ x ∣ \lim_{x \to 0^-}\frac{sinx}{\mid x \mid} limx→0−∣x∣sinx | − 1 -1 −1 |
lim x → + ∞ a r c t a n x \lim_{x \to +\infty} arctanx limx→+∞arctanx | π 2 \frac{\pi}{2} 2π |
lim x → − ∞ a r c t a n x \lim_{x \to -\infty} arctanx limx→−∞arctanx | − π 2 -\frac{\pi}{2} −2π |
lim x → 0 + [ x ] \lim_{x \to 0^+} [x] limx→0+[x] | 0 0 0 |
lim x → 0 − [ x ] \lim_{x \to 0^-} [x] limx→0−[x] | − 1 -1 −1 |
三、微分学
1.基本求导公式
( x α ) ′ = α x α − 1 {(x^\alpha )}' =\alpha x^{\alpha -1} (xα)′=αxα−1
( α x ) ′ = α x l n α {(\alpha^x )}' = \alpha ^xln\alpha (αx)′=αxlnα
( log α x ) ′ = 1 x l n α (\log_{\alpha }{x})' =\frac{1}{xln\alpha} (logαx)′=xlnα1
( log ∣ x ∣ ) ′ = 1 x (\log{\mid x\mid})' =\frac{1}{x} (log∣x∣)′=x1
( s i n x ) ′ = c o s x (sinx)'=cosx (sinx)′=cosx
( c o s x ) ′ = − s i n x (cosx)'=-sinx (cosx)′=−sinx
( t a n x ) ′ = s e c 2 x (tanx)'=sec^2x (tanx)′=sec2x
( a r c s i n x ) ′ = 1 1 − x 2 (arcsinx)'=\frac{1}{\sqrt{1-x^2} } (arcsinx)′=1−x21
( a r c c o s x ) ′ = − 1 1 − x 2 (arccosx)'=-\frac{1}{\sqrt{1-x^2} } (arccosx)′=−1−x21
( a r c t a n x ) ′ = 1 1 + x 2 (arctanx)'=\frac{1}{1+x^2} (arctanx)′=1+x21
( a r c c o t x ) ′ = − 1 1 + x 2 (arccotx)'=-\frac{1}{1+x^2} (arccotx)′=−1+x21
( s e c x ) ′ = s e c x t a n x (secx)'=secx tanx (secx)′=secxtanx
[ l n ( x + x 2 + 1 ) ] ′ = 1 x 2 + 1 {[ln(x+\sqrt{x^2+1})]}'=\frac{1}{\sqrt{x^2+1} } [ln(x+x2+1)]′=x2+11
[ l n ( x + x 2 − 1 ) ] ′ = 1 x 2 − 1 {[ln(x+\sqrt{x^2-1})]}'=\frac{1}{\sqrt{x^2-1} } [ln(x+x2−1)]′=x2−11
[ l n ( x + x 2 + a ) ] ′ = 1 x 2 + a 2 {[ln(x+\sqrt{x^2+a})]}'=\frac{1}{\sqrt{x^2+a^2} } [ln(x+x2+a)]′=x2+a21
2.微分学推广公式\结论
(1)导数定义推论
当 lim x → x 0 f ( x ) x − x 0 = A 当\lim_{x \to x_0} \frac{f(x)}{x-x_0} =A 当limx→x0x−x0f(x)=A,且 f ( x ) 在 x = x 0 f(x)在x=x_0 f(x)在x=x0处连续,则有 f ( x 0 ) = 0 f(x_0)=0 f(x0)=0, f ′ ( x 0 ) = A . f'(x_0)=A. f′(x0)=A.
(2)导函数保号性
y = f ( x ) 可 导 , f ′ ( x ) ≠ 0 , 则 f ′ ( x ) 必 恒 正 或 恒 负 . y=f(x)可导,f'(x)\ne 0,则f'(x)必恒正或恒负. y=f(x)可导,f′(x)=0,则f′(x)必恒正或恒负.
(3)反函数的二阶导数
y x = 1 x y y_x= \frac{1}{x_y} yx=xy1
y x x = − x y y ′ ′ ( x y ′ ) 3 y_{xx}= \frac{-x_{yy}''}{(x_y')^3} yxx=(xy′)3−xyy′′
3.多元函数微分
(1)可微定义
lim △ x → 0 , △ y → 0 △ z − ( A △ x + B △ y ) ( △ x ) 2 + ( △ y ) 2 \lim_{\bigtriangleup x \to 0,\bigtriangleup y \to 0} \frac{\bigtriangleup z-(A\bigtriangleup x+B\bigtriangleup y)}{\sqrt{(\bigtriangleup x)^2+(\bigtriangleup y)^2} } △x→0,△y→0lim(△x)2+(△y)2△z−(A△x+B△y)
(2)全微分
d z = ∂ z ∂ x d x + ∂ z ∂ y d y dz=\frac{\partial z }{\partial x } dx+\frac{\partial z }{\partial y } dy dz=∂x∂zdx+∂y∂zdy
(3)隐函数存在定理
d y d x = − F x ′ F y ′ \frac{dy }{dx }=-\frac{F'_x }{F'_y } dxdy=−Fy′Fx′
(4)二阶偏导数
d 2 y d x 2 = − F x x ′ ′ F y ′ 2 − 2 F x ′ F y ′ F x y ′ ′ + F x ′ 2 F y y ′ ′ F y ′ 3 \frac{d^2y}{dx^2}=-\frac{F''_{xx}F_y'^2-2F'_xF'_yF''_{xy}+F_x'^2F''_{yy}}{F'^3_y} dx2d2y=−Fy′3Fxx′′Fy′2−2Fx′Fy′Fxy′′+Fx′2Fyy′′
四、无穷级数
1.泰勒公式
(1)拉格朗日余项, ξ ∈ [ x , x 0 ] \xi \in [x,x_0] ξ∈[x,x0]
f ( x ) = f ( x 0 ) + f ′ ( x 0 ) ( x − x 0 ) + ⋯ + 1 n ! f ( n ) ( x 0 ) ( x − x 0 ) n + 1 ( n + 1 ) ! f ( n + 1 ) ( ξ ) ( x − x 0 ) ( n + 1 ) f(x)=f(x_0)+f'(x_0)(x-x_0)+\dots +\frac{1}{n!}f^{(n)}(x_0)(x-x_0)^n+\frac{1}{(n+1)!}f^{(n+1)}(\xi )(x-x_0)^{(n+1)} f(x)=f(x0)+f′(x0)(x−x0)+⋯+n!1f(n)(x0)(x−x0)n+(n+1)!1f(n+1)(ξ)(x−x0)(n+1)
(2)配亚诺余项
f ( x ) = f ( x 0 ) + f ′ ( x 0 ) ( x − x 0 ) + 1 2 ! f ′ ′ ( x 0 ) ( x − x 0 ) 2 + ⋯ + 1 n ! f ( n ) ( x 0 ) ( x − x 0 ) n + o ( ( x − x 0 ) n ) f(x)=f(x_0)+f'(x_0)(x-x_0)+\frac{1}{2!}f''(x_0)(x-x_0)^2+\dots +\frac{1}{n!}f^{(n)}(x_0)(x-x_0)^n+o((x-x_0)^n) f(x)=f(x0)+f′(x0)(x−x0)+2!1f′′(x0)(x−x0)2+⋯+n!1f(n)(x0)(x−x0)n+o((x−x0)n)
(3)麦克劳林公式 ( x 0 = 0 ) (x_0=0) (x0=0),拉格朗日余项, ξ ∈ [ 0 , x ] \xi \in [0,x] ξ∈[0,x]
f ( x ) = f ( 0 ) + f ′ ( 0 ) x + ⋯ + f ( n ) ( 0 ) n ! x n + f ( n + 1 ) ( ξ ) ( n + 1 ) ! x n + 1 f(x)=f(0)+f'(0)x+\dots +\frac{f^{(n)}(0)}{n!}x^n+\frac{f^{(n+1)}(\xi )}{(n+1)!}x^{n+1} f(x)=f(0)+f′(0)x+⋯+n!f(n)(0)xn+(n+1)!f(n+1)(ξ)xn+1
(3)麦克劳林公式 ( x 0 = 0 ) (x_0=0) (x0=0),配亚诺余项
f ( x ) = f ( 0 ) + f ′ ( 0 ) x + f ′ ′ ( 0 ) 2 ! x 2 + ⋯ + f ( n ) ( 0 ) n ! x n + o ( x n ) f(x)=f(0)+f'(0)x+\frac{f''(0)}{2!}x^2+\dots +\frac{f^{(n)}(0)}{n!}x^n+o(x^n) f(x)=f(0)+f′(0)x+2!f′′(0)x2+⋯+n!f(n)(0)xn+o(xn)
2.级数敛散性判别法
u n u_n un为数列通项, s n = ∑ n = 0 ∞ u n s_n=\sum_{n=0}^\infty u_n sn=∑n=0∞un为数列和,则有:
适用范围 | 方法 | 内容 |
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正项级数 | 收敛原则(定义) | lim n → ∞ S n = A \lim_{n \to \infty} S_n=A limn→∞Sn=A,级数收敛; lim n → ∞ S n = + ∞ \lim_{n \to \infty} S_n=+\infty limn→∞Sn=+∞,级数发散 |
正项级数 | 比较判别法 | 两个正项级数,大的收敛小的必收敛;小的发散大的必发散 |
正项级数 | 比较判别法推论 | 两个正项级数 ∑ n = 1 ∞ u n , ∑ n = 1 ∞ v n \sum_{n=1}^{\infty} u_n,\sum_{n=1}^{\infty} v_n ∑n=1∞un,∑n=1∞vn且 ≠ 0 \ne 0 =0, lim n → ∞ u n v n = A \lim_{n \to \infty} \frac{u_n}{v_n}=A limn→∞vnun=A, 0 < A < + ∞ 0<A<+ \infty 0<A<+∞,两个级数同敛散。 |
正项级数 | 比值判别法 | 级数 ∑ n = 1 ∞ u n \sum_{n=1}^{\infty} u_n ∑n=1∞un, lim n → ∞ u n + 1 u n = A \lim_{n \to \infty} \frac{u_{n+1}}{u_n}=A limn→∞unun+1=A,若 A < 1 A<1 A<1级数收敛;若 A > 1 A>1 A>1级数发散,若若 A = 1 A=1 A=1该法失效 |
正项级数 | 根值判别法 | 级数 ∑ n = 1 ∞ u n \sum_{n=1}^{\infty} u_n ∑n=1∞un, lim n → ∞ u n n = A \lim_{n \to \infty} \sqrt[n]{u_n}=A limn→∞nun=A,若 A < 1 A<1 A<1级数收敛;若 A > 1 A>1 A>1级数发散,若若 A = 1 A=1 A=1该法失效 |
正项级数 | 积分判别法 | 级数 ∑ n = 1 ∞ u n \sum_{n=1}^{\infty} u_n ∑n=1∞un, f ( x ) f(x) f(x)在 [ N , + ∞ ) [N,+\infty) [N,+∞)上是连续、非负、单调减少, u n = f ( n ) u_n=f(n) un=f(n),则 ∑ n = 1 ∞ u n \sum_{n=1}^{\infty} u_n ∑n=1∞un与 ∫ N ∞ f ( x ) d x \int_{N}^{\infty } f(x)dx ∫N∞f(x)dx同敛散性 |
交错级数 | 莱布尼茨判别法 | 级数 ∑ n = 1 ∞ ( − 1 ) n − 1 u n , u n > 0 , u n \sum_{n=1}^{\infty} {(-1)^{n-1}u_n},u_n>0,u_n ∑n=1∞(−1)n−1un,un>0,un单调不增, lim n → ∞ = 0 \lim_{n \to \infty} =0 limn→∞=0,则交错级数收敛 |
任意级数 | 绝对收敛 | 若级数 ∑ n = 1 ∞ ∣ u n ∣ \sum_{n=1}^{\infty} {\mid u_n\mid } ∑n=1∞∣un∣收敛,则称 ∑ n = 1 ∞ u n \sum_{n=1}^{\infty} {u_n} ∑n=1∞un绝对收敛 |
任意级数 | 条件收敛 | 若级数 ∑ n = 1 ∞ ∣ u n ∣ \sum_{n=1}^{\infty} {\mid u_n\mid } ∑n=1∞∣un∣发散,级数 ∑ n = 1 ∞ u n \sum_{n=1}^{\infty} {u_n } ∑n=1∞un收敛,则称 ∑ n = 1 ∞ u n \sum_{n=1}^{\infty} {u_n} ∑n=1∞un绝对收敛 |
3.常见的级数敛散性
(1)具体级数的敛散性
级数 | 敛散性 |
---|---|
(几何级数/等比级数) ∑ n = 1 ∞ a q n − 1 \sum_{n=1}^{\infty} aq^{n-1} ∑n=1∞aqn−1 | ∣ q ∣ < 1 , 收 敛 \mid q\mid <1,收敛 ∣q∣<1,收敛, ∣ q ∣ ≥ 1 , 发 散 \mid q\mid \ge 1,发散 ∣q∣≥1,发散 |
(p级数) ∑ n = 1 ∞ 1 x p \sum_{n=1}^{\infty} \frac{1}{x^p} ∑n=1∞xp1 | ∣ p ∣ > 1 , 收 敛 \mid p\mid >1,收敛 ∣p∣>1,收敛, ∣ p ∣ ≤ 1 , 发 散 \mid p\mid \le 1,发散 ∣p∣≤1,发散 |
(p积分) ∫ 1 + ∞ 1 x p d x \int_{1}^{+\infty } \frac{1}{x^p} dx ∫1+∞xp1dx | ∣ p ∣ > 1 , 收 敛 \mid p\mid >1,收敛 ∣p∣>1,收敛, ∣ p ∣ ≤ 1 , 发 散 \mid p\mid \le 1,发散 ∣p∣≤1,发散 |
∫ 0 1 1 x p d x \int_{0}^{1 } \frac{1}{x^p} dx ∫01xp1dx | 0 < p < 1 , 收 敛 0< p <1,收敛 0<p<1,收敛, p ≥ 1 , 发 散 p\ge 1,发散 p≥1,发散 |
(广义p级数) ∑ n = 2 ∞ 1 x l n p x \sum_{n=2}^{\infty} \frac{1}{xln^px} ∑n=2∞xlnpx1 | ∣ p ∣ > 1 , 收 敛 \mid p\mid >1,收敛 ∣p∣>1,收敛, ∣ p ∣ ≤ 1 , 发 散 \mid p\mid \le 1,发散 ∣p∣≤1,发散 |
(广义p积分) ∫ 2 + ∞ 1 x l n p x d x \int_{2}^{+\infty } \frac{1}{xln^px} dx ∫2+∞xlnpx1dx | ∣ p ∣ > 1 , 收 敛 \mid p\mid >1,收敛 ∣p∣>1,收敛, ∣ p ∣ ≤ 1 , 发 散 \mid p\mid \le 1,发散 ∣p∣≤1,发散 |
(调和级数) ∑ n = 1 ∞ 1 x \sum_{n=1}^{\infty} \frac{1}{x} ∑n=1∞x1 | 发散 |
∑ n = 1 ∞ ( − 1 ) n 1 x \sum_{n=1}^{\infty} (-1)^n\frac{1}{x} ∑n=1∞(−1)nx1 | 收敛 |
∑ n = 1 ∞ ( − 1 ) n 1 x \sum_{n=1}^{\infty} (-1)^n\frac{1}{\sqrt{x}} ∑n=1∞(−1)nx1 | 收敛 |
(2)抽象级数的判敛散问题
∑ n = 1 ∞ u n \sum_{n=1}^{\infty} u_n ∑n=1∞un, ∑ n = 1 ∞ v n \sum_{n=1}^{\infty} v_n ∑n=1∞vn, ∑ n = 1 ∞ w n \sum_{n=1}^{\infty} w_n ∑n=1∞wn均是任意项级数,则有:
条件 | 结论 |
---|---|
a , b , c a,b,c a,b,c为非零常数, a u n + b v n + c w n = 0 au_n+bv_n+cw_n=0 aun+bvn+cwn=0 | ∑ n = 1 ∞ u n \sum_{n=1}^{\infty} u_n ∑n=1∞un, ∑ n = 1 ∞ v n \sum_{n=1}^{\infty} v_n ∑n=1∞vn, ∑ n = 1 ∞ w n \sum_{n=1}^{\infty} w_n ∑n=1∞wn中有两个级数收敛,第三个必收敛 |
∑ n = 1 ∞ ∣ u n ∣ \sum_{n=1}^{\infty} \mid u_n\mid ∑n=1∞∣un∣收敛 | ∑ n = 1 ∞ u n \sum_{n=1}^{\infty} u_n ∑n=1∞un收敛 |
∑ n = 1 ∞ u n \sum_{n=1}^{\infty} u_n ∑n=1∞un发散 | ∑ n = 1 ∞ ∣ u n ∣ \sum_{n=1}^{\infty} \mid u_n\mid ∑n=1∞∣un∣发散 |
∑ n = 1 ∞ u n \sum_{n=1}^{\infty} u_n ∑n=1∞un收敛 | ∑ n = 1 ∞ ∣ u n ∣ \sum_{n=1}^{\infty} \mid u_n\mid ∑n=1∞∣un∣不定 |
∑ n = 1 ∞ u n 2 \sum_{n=1}^{\infty} u^2_n ∑n=1∞un2收敛 | ∑ n = 1 ∞ u n n \sum_{n=1}^{\infty} \frac{u_n}{n} ∑n=1∞nun绝对收敛 |
∑ n = 1 ∞ u n \sum_{n=1}^{\infty} u_n ∑n=1∞un收敛 | ∑ n = 1 ∞ u n 2 \sum_{n=1}^{\infty} u^2_n ∑n=1∞un2不定 |
∑ n = 1 ∞ u n \sum_{n=1}^{\infty} u_n ∑n=1∞un收敛 | ∑ n = 1 ∞ ( − 1 ) n u n \sum_{n=1}^{\infty} (-1)^nu_n ∑n=1∞(−1)nun不定 |
∑ n = 1 ∞ u n \sum_{n=1}^{\infty} u_n ∑n=1∞un收敛 | ∑ n = 1 ∞ ( − 1 ) n u n n \sum_{n=1}^{\infty} (-1)^n \frac{u_n}{n} ∑n=1∞(−1)nnun不定 |
∑ n = 1 ∞ u n \sum_{n=1}^{\infty} u_n ∑n=1∞un收敛 | ∑ n = 1 ∞ u 2 n \sum_{n=1}^{\infty} u_{2n} ∑n=1∞u2n(偶数项)不定, ∑ n = 1 ∞ u 2 n − 1 \sum_{n=1}^{\infty} u_{2n-1} ∑n=1∞u2n−1(奇数项)不定 |
∑ n = 1 ∞ u n \sum_{n=1}^{\infty} u_n ∑n=1∞un收敛 | ∑ n = 1 ∞ ( u 2 n − 1 + u 2 n ) \sum_{n=1}^{\infty}( u_{2n-1}+u_{2n}) ∑n=1∞(u2n−1+u2n)收敛 |
∑ n = 1 ∞ u n \sum_{n=1}^{\infty} u_n ∑n=1∞un收敛 | ∑ n = 1 ∞ ( u 2 n − 1 − u 2 n ) \sum_{n=1}^{\infty}( u_{2n-1}-u_{2n}) ∑n=1∞(u2n−1−u2n)不定 |
∑ n = 1 ∞ u n \sum_{n=1}^{\infty} u_n ∑n=1∞un收敛 | ∑ n = 1 ∞ ( u n + u n + 1 ) \sum_{n=1}^{\infty}( u_{n}+u_{n+1}) ∑n=1∞(un+un+1)收敛 |
∑ n = 1 ∞ u n \sum_{n=1}^{\infty} u_n ∑n=1∞un收敛 | ∑ n = 1 ∞ ( u n − u n + 1 ) \sum_{n=1}^{\infty}( u_{n}-u_{n+1}) ∑n=1∞(un−un+1)收敛 |
∑ n = 1 ∞ u n \sum_{n=1}^{\infty} u_n ∑n=1∞un收敛 | ∑ n = 1 ∞ u n + ∑ n = 1 ∞ u n + 1 \sum_{n=1}^{\infty}u_{n}+\sum_{n=1}^{\infty}u_{n+1} ∑n=1∞un+∑n=1∞un+1收敛 |
∑ n = 1 ∞ u n \sum_{n=1}^{\infty} u_n ∑n=1∞un收敛 | ∑ n = 1 ∞ u n − ∑ n = 1 ∞ u n + 1 \sum_{n=1}^{\infty}u_{n}-\sum_{n=1}^{\infty}u_{n+1} ∑n=1∞un−∑n=1∞un+1收敛 |
∑ n = 1 ∞ u n \sum_{n=1}^{\infty} u_n ∑n=1∞un收敛 | ∑ n = 1 ∞ u n u n + 1 \sum_{n=1}^{\infty} u_{n}u_{n+1} ∑n=1∞unun+1不定 |
4.常见函数的幂级数展开式
展开式 | 收敛域 |
---|---|
e x = ∑ n = 0 ∞ x n n ! = 1 + x + x 2 2 ! + ⋯ + x n n ! + … e^x=\sum_{n=0}^{\infty } \frac{x^n}{n!} =1+x+\frac{x^2}{2!}+\dots +\frac{x^n}{n!}+\dots ex=∑n=0∞n!xn=1+x+2!x2+⋯+n!xn+… | − ∞ < x < + ∞ - \infty <x<+\infty −∞<x<+∞ |
1 x + 1 = ∑ n = 0 ∞ ( − 1 ) n x n = 1 − x + x 2 − x 3 + ⋯ + ( − 1 ) n x n + … \frac{1}{x+1} =\sum_{n=0}^{\infty } (-1)^nx^n =1-x+x^2-x^3+\dots +(-1)^nx^n+\dots x+11=∑n=0∞(−1)nxn=1−x+x2−x3+⋯+(−1)nxn+… | − 1 < x < 1 - 1 <x<1 −1<x<1 |
1 x − 1 = ∑ n = 0 ∞ x n = 1 + x + x 2 + ⋯ + x n + … \frac{1}{x-1} =\sum_{n=0}^{\infty }x^n =1+x+x^2+\dots +x^n+\dots x−11=∑n=0∞xn=1+x+x2+⋯+xn+… | − 1 < x < 1 - 1 <x<1 −1<x<1 |
l n ( 1 + x ) = ∑ n = 0 ∞ ( − 1 ) n − 1 x n n = x − x 2 2 + ⋯ + ( − 1 ) n − 1 x n n + … ln(1+x) =\sum_{n=0}^{\infty } (-1)^{n-1}\frac{x^n}{n} =x-\frac{x^2}{2} +\dots +(-1)^{n-1}\frac{x^n}{n}+\dots ln(1+x)=∑n=0∞(−1)n−1nxn=x−2x2+⋯+(−1)n−1nxn+… | − 1 < x ≤ 1 - 1 <x\le 1 −1<x≤1 |
( 1 + x ) α = 1 + α x + α ( α − 1 ) 2 ! x 2 + … α ( α − 1 ) … ( α − n + 1 ) n ! x n + … (1+x)^\alpha =1+\alpha x+\frac{\alpha (\alpha -1)}{2!} x^2+\dots \frac{\alpha (\alpha -1)\dots (\alpha-n+1)}{n!}x^n+\dots (1+x)α=1+αx+2!α(α−1)x2+…n!α(α−1)…(α−n+1)xn+… | 当 α ≤ − 1 , x ∈ ( − 1 , 1 ) 当\alpha \le -1,x\in (-1,1) 当α≤−1,x∈(−1,1), 当 − 1 < α < 0 , x ∈ ( − 1 , 1 ] 当-1<\alpha <0,x\in (-1,1] 当−1<α<0,x∈(−1,1], 当 α > 0 , α ∉ N + , x ∈ [ − 1 , 1 ] 当\alpha >0,\alpha\notin N_+,x\in [-1,1] 当α>0,α∈/N+,x∈[−1,1], 当 α > 0 , α ∈ N + , x ∈ R 当\alpha >0,\alpha\in N_+,x\in R 当α>0,α∈N+,x∈R |
− l n ( 1 − x ) = ∑ n = 1 ∞ x n n -ln(1-x) =\sum_{n=1}^{\infty } \frac{x^n}{n} −ln(1−x)=∑n=1∞nxn | − 1 ≤ x < 1 - 1 \le x< 1 −1≤x<1 |
1 ( 1 − x ) 2 = ∑ n = 1 ∞ n x n − 1 \frac{1}{(1-x)^2} =\sum_{n=1}^{\infty } nx^{n-1} (1−x)21=∑n=1∞nxn−1 | − 1 < x < 1 - 1 < x< 1 −1<x<1 |
s i n x = ∑ n = 0 ∞ ( − 1 ) n x 2 n + 1 ( 2 n + 1 ) ! sinx =\sum_{n=0}^{\infty } (-1)^{n}\frac{x^{2n+1}}{(2n+1)!} sinx=∑n=0∞(−1)n(2n+1)!x2n+1 | − ∞ < x < + ∞ - \infty <x< +\infty −∞<x<+∞ |
c o s x = ∑ n = 0 ∞ ( − 1 ) n x 2 n ( 2 n ) ! cosx =\sum_{n=0}^{\infty } (-1)^{n}\frac{x^{2n}}{(2n)!} cosx=∑n=0∞(−1)n(2n)!x2n | − ∞ < x < + ∞ - \infty <x< +\infty −∞<x<+∞ |
5.幂级数运算法则
运算法则 | 公式 |
---|---|
通项,下标一起变 | ∑ n = k ∞ a n x n = ∑ n = k + l ∞ a n − l x n − l \sum_{n=k}^{\infty}a_nx^n=\sum_{n=k+l}^{\infty}a_{n-l}x^{n-l} ∑n=k∞anxn=∑n=k+l∞an−lxn−l |
通项不变,下标变 | ∑ n = k ∞ a n x n = a k x k + a k + 1 x k + 1 + ⋯ + a k + l − 1 x k + l − 1 + ∑ n = k + l ∞ a n x n \sum_{n=k}^{\infty}a_nx^n=a_kx^k+a_{k+1}x^{k+1}+ \dots+a_{k+l-1}x^{k+l-1}+\sum_{n=k+l}^{\infty}a_nx^n ∑n=k∞anxn=akxk+ak+1xk+1+⋯+ak+l−1xk+l−1+∑n=k+l∞anxn |
通项变,下标不变 | ∑ n = k ∞ a n x n = x l ∑ n = k ∞ a n x n − l \sum_{n=k}^{\infty}a_nx^n=x^l\sum_{n=k}^{\infty}a_nx^{n-l} ∑n=k∞anxn=xl∑n=k∞anxn−l |
五、中值定理
1.中值定理
定理 | 条件 | 结论 |
---|---|---|
有界与最值定理 | f ( x ) f(x) f(x)在 [ a , b ] [a,b] [a,b]上连续, m ≤ f ( x ) ≤ M m\le f(x)\le M m≤f(x)≤M | m , M m,M m,M为 f ( x ) f(x) f(x)在 [ a , b ] [a,b] [a,b]上的最小值与最大值 |
介值定理 | f ( x ) f(x) f(x)在 [ a , b ] [a,b] [a,b]上连续, m ≤ μ ≤ M m\le \mu \le M m≤μ≤M | 存在 ξ ∈ [ a , b ] , f ( ξ ) = μ \xi \in [a,b],f(\xi)=\mu ξ∈[a,b],f(ξ)=μ |
(离散的)平均值定理 | f ( x ) f(x) f(x)在 [ a , b ] [a,b] [a,b]上连续, a < x 1 < x 2 < ⋯ < x n < b a<x_1<x_2<\dots<x_n<b a<x1<x2<⋯<xn<b | 至少存在一点 ξ ∈ [ x 1 , x n ] , f ( ξ ) = f ( x 1 ) + f ( x 2 ) + ⋯ + f ( x n ) n \xi \in [x_1,x_n],f(\xi)=\frac{f(x_1)+f(x_2)+\dots+f(x_n)}{n} ξ∈[x1,xn],f(ξ)=nf(x1)+f(x2)+⋯+f(xn) |
零点定理 | f ( x ) f(x) f(x)在 [ a , b ] [a,b] [a,b]上连续, f ( a ) ⋅ f ( b ) < 0 f(a)\cdot f(b)<0 f(a)⋅f(b)<0 | 存在 ξ ∈ ( a , b ) , f ( ξ ) = 0 \xi \in (a,b),f(\xi)=0 ξ∈(a,b),f(ξ)=0 |
费马定理 | f ( x ) f(x) f(x)在 x 0 x_0 x0处可导, f ( x ) f(x) f(x)在 x 0 x_0 x0处取极值 | f ′ ( x 0 ) = 0 f'(x_0)=0 f′(x0)=0 |
罗尔定理 | f ( x ) f(x) f(x)在 [ a , b ] [a,b] [a,b]上连续, f ( x ) f(x) f(x)在 ( a , b ) (a,b) (a,b)上可导, f ( a ) = f ( b ) f(a)=f(b) f(a)=f(b) | 存在 ξ ∈ ( a , b ) , f ( ξ ) = 0 \xi \in (a,b),f(\xi)=0 ξ∈(a,b),f(ξ)=0 |
拉格朗日中值定理 | f ( x ) f(x) f(x)在 [ a , b ] [a,b] [a,b]上连续, f ( x ) f(x) f(x)在 ( a , b ) (a,b) (a,b)上可导 | 存在 ξ ∈ ( a , b ) , f ( b ) − f ( a ) = f ′ ( ξ ) ( b − a ) \xi \in (a,b),f(b)-f(a)=f'(\xi )(b-a) ξ∈(a,b),f(b)−f(a)=f′(ξ)(b−a) |
柯西中值定理 | f ( x ) , g ( x ) f(x),g(x) f(x),g(x)在 [ a , b ] [a,b] [a,b]上连续, f ( x ) , g ( x ) f(x),g(x) f(x),g(x)在 ( a , b ) (a,b) (a,b)上可导, g ′ ( x ) ≠ 0 g'(x)\ne 0 g′(x)=0 | 存在 ξ ∈ ( a , b ) , f ( b ) − f ( a ) g ( b ) − g ( a ) = f ′ ( ξ ) g ′ ( ξ ) \xi \in (a,b),\frac{f(b)-f(a)}{g(b)-g(a)} =\frac{f'(\xi)}{g'(\xi)} ξ∈(a,b),g(b)−g(a)f(b)−f(a)=g′(ξ)f′(ξ) |
积分中值定理 | f ( x ) f(x) f(x)在 [ a , b ] [a,b] [a,b]上连续, f ( x ) f(x) f(x)在 ( a , b ) (a,b) (a,b)上可导 | 存在 ξ ∈ [ a , b ] , ∫ a b f ( x ) d x = f ( ξ ) ( b − a ) \xi \in [a,b],\int_{a}^{b} f(x)dx=f(\xi)(b-a) ξ∈[a,b],∫abf(x)dx=f(ξ)(b−a) |
导数零点定理 | f ( x ) f(x) f(x)在 [ a , b ] [a,b] [a,b]上可导, f + ′ ( a ) ⋅ f − ′ ( b ) < 0 f'_+(a) \cdot f'_-(b)<0 f+′(a)⋅f−′(b)<0 | 存在 ξ ∈ ( a , b ) , f ( ξ ) = 0 \xi \in (a,b),f(\xi)=0 ξ∈(a,b),f(ξ)=0 |
2.常见辅助函数的构造方法
u v ′ + u ′ v = ( u v ) ′ uv'+u'v=(uv)' uv′+u′v=(uv)′
f ( x ) f ′ ( x ) = 1 2 [ f 2 ( x ) ] ′ f(x)f'(x)=\frac{1 }{2} [f^2(x)]' f(x)f′(x)=21[f2(x)]′
[ f ′ ( x ) ] 2 + f ( x ) f ′ ′ ( x ) = [ f ( x ) ⋅ f ′ ( x ) ] [f'(x)]^2+f(x)f''(x)=[f(x) \cdot f'(x)] [f′(x)]2+f(x)f′′(x)=[f(x)⋅f′(x)]
[ f ′ ( x ) + f ( x ) φ ′ ( x ) ] e φ ( x ) = [ f ( x ) e φ ( x ) ] ′ [f'(x)+f(x)\varphi'(x)]e^{\varphi(x)}=[f(x)e^{\varphi(x)}]' [f′(x)+f(x)φ′(x)]eφ(x)=[f(x)eφ(x)]′
对于上式,
当 φ ( x ) = x \varphi(x)=x φ(x)=x时, f ( x ) + f ′ ( x ) = [ f ( x ) e x ] ′ e x f(x)+f'(x)=\frac{[f(x)e^x]'}{e^x} f(x)+f′(x)=ex[f(x)ex]′
当 φ ( x ) = − x \varphi(x)=-x φ(x)=−x时, f ( x ) − f ′ ( x ) = [ f ( x ) e − x ] ′ e − x f(x)-f'(x)=\frac{[f(x)e^{-x}]'}{e^{-x}} f(x)−f′(x)=e−x[f(x)e−x]′
当 φ ( x ) = k x \varphi(x)=kx φ(x)=kx时, f ( x ) + k f ′ ( x ) = [ f ( x ) e k x ] ′ e k x f(x)+kf'(x)=\frac{[f(x)e^{kx}]'}{e^{kx}} f(x)+kf′(x)=ekx[f(x)ekx]′