柯西收敛原理演示

1from manim import *2import numpy as np3 4config.tex_template = TexTemplateLibrary.ctex5config.tex_template.add_to_preamble(r"\setCJKmainfont{STSong}")6 7class CauchyConvergence(Scene):8    def construct(self):9        # 创建标题10        title = Text("柯西收敛原理演示", font="STSong", font_size=48)11        12        self.play(Write(title))13        self.wait(1)14        self.play(FadeOut(title))15 16        # 创建坐标系17        axes = Axes(18            x_range=[0, 10, 1],19            y_range=[0, 1.2, 0.2],20            axis_config={"include_tip": True},21            x_length=10,22            y_length=623        )24        25        x_label = axes.get_x_axis_label("n")26        y_label = axes.get_y_axis_label("a_n")27        28        self.play(Create(axes), Write(x_label), Write(y_label))29 30        # 创建数列点31        def get_points(n_points):32            return [axes.coords_to_point(x, 1/x) for x in range(1, n_points + 1)]33 34        dots = VGroup(*[Dot(point) for point in get_points(10)])35        36        # 显示点37        self.play(Create(dots))38        self.wait(1)39 40        # 演示柯西条件41        # 先创建柯西条件的数学表达式42        cauchy_condition = MathTex(43            r"\forall \varepsilon > 0, \exists N \in \mathbb{N}, ",44            r"\forall n,m > N: |a_n - a_m| < \varepsilon"45        ).scale(0.8)46        # 将表达式移到右上角47        cauchy_condition.to_corner(UR)48        self.play(Write(cauchy_condition))49        50        # 创建说明文字（在循环外创建）51        explanation = Text(52            "N之后任意两项差的绝对值不会超过ε",53            font="STSong",54            color=YELLOW55        ).scale(0.4)56        # 将说明文字固定在上方位置57        explanation.to_edge(UP).shift(DOWN * 2)58 59        # 依次演示不同的ε值60        for i, epsilon in enumerate([0.3, 0.2, 0.1]):61            N = int(1/epsilon) - 162            min_y = 1/(N+5)  # N之后某个点的y值63            max_y = 1/N      # N处的y值64            reference_y = (max_y + min_y) / 265            66            # 创建两条水平线，确保包含曲线，并延长到整个x轴范围67            upper_line = DashedLine(68                axes.coords_to_point(0, max_y + epsilon/4),69                axes.coords_to_point(10, max_y + epsilon/4),70                color=RED71            )72            lower_line = DashedLine(73                axes.coords_to_point(0, min_y - epsilon/4),74                axes.coords_to_point(10, min_y - epsilon/4),75                color=RED76            )77            78            # 添加竖直的距离标注（移到右侧）79            distance_line = Line(80                axes.coords_to_point(9.5, max_y + epsilon/4),81                axes.coords_to_point(9.5, min_y - epsilon/4),82                color=RED83            )84            epsilon_brace = Brace(distance_line, direction=RIGHT, color=RED)85            epsilon_label = MathTex(r"\varepsilon", color=RED).next_to(epsilon_brace, RIGHT)86            87            epsilon_band = VGroup(88                upper_line, lower_line,89                distance_line, epsilon_brace, epsilon_label90            )91            92            self.play(93                Create(upper_line),94                Create(lower_line),95                Create(distance_line),96                Create(epsilon_brace),97                Write(epsilon_label)98            )99            100            # 创建N的垂直线101            N_line = DashedLine(102                axes.coords_to_point(N, 0),103                axes.coords_to_point(N, 1.2),104                color=GREEN105            )106            # 将N的标注移到下方107            N_text = Text("N", font="STSong", color=GREEN).next_to(108                axes.coords_to_point(N, 0),109                DOWN110            )111            112            # 显示N线113            self.play(114                Create(N_line),115                Write(N_text)116            )117 118            # 在N显示后再显示说明文字119            if i == 0:120                self.play(Write(explanation))121            122            self.wait(1.5)123            124            # 清除当前ε的演示内容125            self.play(126                FadeOut(epsilon_band),127                FadeOut(N_line),128                FadeOut(N_text)129            )130 131        # 显示极限存在的结论132        limit_text = VGroup(133            MathTex(r"\lim_{n \to \infty} a_n"),134            Text("存在", font="STSong", color=WHITE).scale(0.8)135        ).arrange(RIGHT, buff=0.2)136        # 将结论放在定理下方137        limit_text.next_to(cauchy_condition, DOWN, buff=0.5)138        self.play(Write(limit_text))139        self.wait(2)140 141def main():142    import os143    os.system("manim -pql cauchy_convergence.py CauchyConvergence")144 145if __name__ == "__main__":146    main()