利用定义证明数列极限不存在

1from manim import *2import numpy as np3 4class EmphasizeLimitDefinition(Scene):5    def construct(self):6        # Title7        title = Text("证明数列极限不存在", font="SimSun", font_size=36)8        title.to_edge(UP)9        self.play(Write(title))10        self.wait(1)11        12        # Show sequence a_n = (-1)^n13        sequence_def = MathTex(r"a_n = (-1)^n", font_size=40)14        sequence_def.next_to(title, DOWN, buff=0.3)15        self.play(Write(sequence_def))16        self.wait(2)17        18        # Create axes higher up19        axes = Axes(20            x_range=[0, 30, 5],21            y_range=[-2, 2, 1],22            axis_config={"include_tip": True},23            x_length=10,  # 缩小x轴长度24            y_length=5    # 缩小y轴长度25        )26        axes.next_to(sequence_def, DOWN, buff=0)  # 向上移动0.5个单位（减少buff值）27        28        # Axes labels29        x_label = MathTex("n", font_size=30).next_to(axes.x_axis, RIGHT)30        y_label = MathTex("a_n", font_size=30).next_to(axes.y_axis, UP)31        32        # 定义outside_after_10变量在更外层的作用域33        outside_after_10 = VGroup()34        35        self.play(Create(axes), Write(x_label), Write(y_label))36        self.wait(2)37        38        # Plot sequence points39        def seq_func(n):40            return (-1)**n41        42        # Create all points43        points = [axes.c2p(n, seq_func(n)) for n in range(1, 31)]44        dots = VGroup(*[Dot(point, color=BLUE, radius=0.07) for point in points])45        46        # Show all points at once47        self.play(Create(dots), run_time=2)48        self.wait(2)49        50        # Try to show convergence to L = 151        limit_line_1 = DashedLine(52            axes.c2p(0, 1),53            axes.c2p(30, 1),54            color=YELLOW55        )56        limit_label_1 = MathTex("L = 1", color=YELLOW, font_size=30).next_to(limit_line_1, RIGHT, buff=0.2)57        58        self.play(Create(limit_line_1), Write(limit_label_1))59        self.wait(1)60        61        # Show epsilon band around L = 162        epsilon_val = 0.563        epsilon_band_upper = DashedLine(64            axes.c2p(0, 1 + epsilon_val),65            axes.c2p(30, 1 + epsilon_val),66            color=YELLOW67        )68        epsilon_band_lower = DashedLine(69            axes.c2p(0, 1 - epsilon_val),70            axes.c2p(30, 1 - epsilon_val),71            color=YELLOW72        )73        74        epsilon_brace = BraceBetweenPoints(75            axes.c2p(25, 1 - epsilon_val),76            axes.c2p(25, 1 + epsilon_val),77            color=YELLOW78        )79        epsilon_label = MathTex("2\\varepsilon = 1.0", color=YELLOW, font_size=24).next_to(epsilon_brace, RIGHT, buff=0.1).shift(DOWN*0.2)80        81        self.play(82            Create(epsilon_band_upper),83            Create(epsilon_band_lower),84            GrowFromCenter(epsilon_brace),85            Write(epsilon_label)86        )87        self.wait(2)88        89        # Highlight points outside epsilon band (odd terms at -1)90        odd_points = VGroup(*[dots[i] for i in range(1, len(dots), 2)])  # Points at -191        self.play(odd_points.animate.set_color(RED), run_time=1.5)92        self.wait(1)93        94        # Emphasize the key concept: for any N, there exists n > N outside the band95        concept_text = Text("关键概念：对于任意N，总存在n>N使得点在带外", font="SimSun", font_size=24)96        concept_text.next_to(axes, DOWN, buff=0.3).shift(UP*0.8)97        self.play(Write(concept_text))98        self.wait(2)99        100        # Demonstrate this concept with different values of N101        # N = 5102        n_line_5 = DashedLine(103            axes.c2p(5, -2),104            axes.c2p(5, 2),105            color=GREEN106        )107        n_label_5 = MathTex("N = 5", color=GREEN, font_size=24).next_to(n_line_5, UP, buff=0.1)108        109        self.play(Create(n_line_5), Write(n_label_5))110        self.wait(1)111        112        # Highlight points after N=5 that are outside the band (points at -1)113        outside_after_5 = VGroup()114        for n in range(5+1, 31):  # n > N (5)115            if n % 2 == 1:  # Odd n values correspond to points at -1116                outside_after_5.add(dots[n-1])  # Subtract 1 to convert n to index117        118        # Add highlight circles around the points119        highlight_circles_5 = VGroup()120        for point in outside_after_5:121            circle = Circle(radius=0.15, color=ORANGE, stroke_width=5).move_to(point)122            highlight_circles_5.add(circle)123        124        self.play(125            outside_after_5.animate.set_color(ORANGE), 126            *[Create(circle) for circle in highlight_circles_5],127            run_time=1.5128        )129        self.wait(1)130        131        # Show that there are points outside the band after N=5132        exists_text_5 = Text("存在n=7,9,11,... > 5使得点在带外", 133                           font="SimSun", font_size=20, color=ORANGE)134        exists_text_5.next_to(axes, DOWN, buff=0.3).shift(UP*1.3)135        self.play(Write(exists_text_5))136        self.wait(2)137        138        # 恢复淡出文字和线条，并恢复点的颜色139        self.play(140            FadeOut(exists_text_5),141            FadeOut(n_line_5),142            FadeOut(n_label_5),143            FadeOut(highlight_circles_5),144            outside_after_5.animate.set_color(BLUE)145        )146        147        # Add demonstration for N = 20148        # N = 20149        n_line_20 = DashedLine(150            axes.c2p(20, -2),151            axes.c2p(20, 2),152            color=RED153        )154        n_label_20 = MathTex("N = 20", color=RED, font_size=24).next_to(n_line_20, UP, buff=0.1)155        156        self.play(Create(n_line_20), Write(n_label_20))157        self.wait(1)158        159        # Highlight points after N=20 that are outside the band (points at -1)160        outside_after_20 = VGroup()161        for n in range(20+1, 31):  # n > N (20)162            if n % 2 == 1:  # Odd n values correspond to points at -1163                outside_after_20.add(dots[n-1])  # Subtract 1 to convert n to index164        165        # Add highlight circles around the points166        highlight_circles_20 = VGroup()167        for point in outside_after_20:168            circle = Circle(radius=0.15, color=ORANGE, stroke_width=5).move_to(point)169            highlight_circles_20.add(circle)170        171        self.play(172            outside_after_20.animate.set_color(ORANGE),173            *[Create(circle) for circle in highlight_circles_20],174            run_time=1.5175        )176        self.wait(1)177        178        # Show that there are points outside the band after N=20179        exists_text_20 = Text("存在n=21,23，... > 20使得点在带外", 180                            font="SimSun", font_size=20, color=ORANGE)181        exists_text_20.next_to(axes, DOWN, buff=0.3).shift(UP*1.3)182        self.play(Write(exists_text_20))183        self.wait(2)184        185        # 恢复淡出文字和线条，但不淡出点186        self.play(187            FadeOut(exists_text_20),188            FadeOut(n_line_20),189            FadeOut(n_label_20),190            FadeOut(highlight_circles_20),191            outside_after_20.animate.set_color(BLUE)192        )193        194        # Fade out first attempt but keep axes and dots195        self.play(196            FadeOut(limit_line_1),197            FadeOut(limit_label_1),198            FadeOut(epsilon_band_upper),199            FadeOut(epsilon_band_lower),200            FadeOut(epsilon_brace),201            FadeOut(epsilon_label),202            FadeOut(concept_text),203            odd_points.animate.set_color(BLUE)204        )205        self.wait(1)206        207        # Try to show convergence to L = -1208        limit_line_neg1 = DashedLine(209            axes.c2p(0, -1),210            axes.c2p(30, -1),211            color=GREEN212        )213        limit_label_neg1 = MathTex("L = -1", color=GREEN, font_size=30).next_to(limit_line_neg1, RIGHT, buff=0.2)214        215        self.play(Create(limit_line_neg1), Write(limit_label_neg1))216        self.wait(1)217        218        # Show epsilon band around L = -1219        epsilon_band_neg1_upper = DashedLine(220            axes.c2p(0, -1 + epsilon_val),221            axes.c2p(30, -1 + epsilon_val),222            color=GREEN223        )224        epsilon_band_neg1_lower = DashedLine(225            axes.c2p(0, -1 - epsilon_val),226            axes.c2p(30, -1 - epsilon_val),227            color=GREEN228        )229        230        epsilon_brace_neg1 = BraceBetweenPoints(231            axes.c2p(25, -1 - epsilon_val),232            axes.c2p(25, -1 + epsilon_val),233            color=GREEN234        )235        epsilon_label_neg1 = MathTex("2\\varepsilon = 1.0", color=GREEN, font_size=24).next_to(epsilon_brace_neg1, RIGHT, buff=0.1).shift(DOWN*0.2)236        237        self.play(238            Create(epsilon_band_neg1_upper),239            Create(epsilon_band_neg1_lower),240            GrowFromCenter(epsilon_brace_neg1),241            Write(epsilon_label_neg1)242        )243        self.wait(2)244        245        # Highlight points outside epsilon band (even terms at +1)246        even_points = VGroup(*[dots[i] for i in range(0, len(dots), 2)])  # Points at +1247        self.play(even_points.animate.set_color(RED), run_time=1.5)248        self.wait(1)249        250        # Emphasize the key concept again251        concept_text_neg1 = Text("关键概念：对于任意N，总存在n>N使得点在带外", font="SimSun", font_size=24)252        concept_text_neg1.next_to(axes, DOWN, buff=0.3).shift(UP*0.8)253        self.play(Write(concept_text_neg1))254        self.wait(2)255        256        # Demonstrate with N = 10257        n_line_10 = DashedLine(258            axes.c2p(10, -2),259            axes.c2p(10, 2),260            color=PURPLE261        )262        n_label_10 = MathTex("N = 10", color=PURPLE, font_size=24).next_to(n_line_10, UP, buff=0.1)263        264        self.play(Create(n_line_10), Write(n_label_10))265        self.wait(1)266        267        # Highlight points after N=10 that are outside the band (points at +1)268        # outside_after_10 = VGroup()  # 已经在外部定义269        for n in range(10+1, 31):  # n > N (10)270            if n % 2 == 0:  # Even n values correspond to points at +1271                outside_after_10.add(dots[n-1])  # Subtract 1 to convert n to index272        273        # Add highlight circles around the points274        highlight_circles_10 = VGroup()275        for point in outside_after_10:276            circle = Circle(radius=0.15, color=ORANGE, stroke_width=5).move_to(point)277            highlight_circles_10.add(circle)278        279        self.play(280            outside_after_10.animate.set_color(ORANGE),281            *[Create(circle) for circle in highlight_circles_10],282            run_time=1.5283        )284        self.wait(1)285        286        # Show that there are points outside the band after N=10287        exists_text_10 = Text("存在n=12,14,16,... > 10使得点在带外", font="SimSun", font_size=20, color=ORANGE)288        exists_text_10.next_to(axes, DOWN, buff=0.3).shift(UP*1.3)289        self.play(Write(exists_text_10))290        self.wait(2)291        292        # 恢复淡出文字和线条，但不淡出点293        self.play(294            FadeOut(exists_text_10),295            FadeOut(n_line_10),296            FadeOut(n_label_10),297            FadeOut(highlight_circles_10),298            outside_after_10.animate.set_color(BLUE)299            # 不再淡出 outside_after_10300        )301        302        # N = 25 (moved from later to demonstrate not converging to -1)303        n_line_25 = DashedLine(304            axes.c2p(25, -2),305            axes.c2p(25, 2),306            color=GREEN307        )308        n_label_25 = MathTex("N = 25", color=GREEN, font_size=24).next_to(n_line_25, UP, buff=0.1)309        310        self.play(Create(n_line_25), Write(n_label_25))311        self.wait(1)312        313        # Highlight points after N=25 that are outside the band (points at +1)314        outside_after_25 = VGroup()315        for n in range(25+1, 31):  # n > N (25)316            if n % 2 == 0:  # Even n values correspond to points at +1317                outside_after_25.add(dots[n-1])  # Subtract 1 to convert n to index318        319        # Add highlight circles around the points320        highlight_circles_25 = VGroup()321        for point in outside_after_25:322            circle = Circle(radius=0.15, color=ORANGE, stroke_width=5).move_to(point)323            highlight_circles_25.add(circle)324        325        self.play(326            outside_after_25.animate.set_color(ORANGE),327            *[Create(circle) for circle in highlight_circles_25],328            run_time=1.5329        )330        self.wait(1)331        332        # Show that there are points outside the band after N=25333        exists_text_25 = Text("存在n=26,28，... > 25使得点在带外", 334                            font="SimSun", font_size=20, color=ORANGE)335        exists_text_25.next_to(axes, DOWN, buff=0.3).shift(UP*1.3)336        self.play(Write(exists_text_25))337        self.wait(2)338        339        # Show conclusion that sequence does not converge to L=-1340        not_converge_neg1 = Text("数列不以-1为极限", font="SimSun", font_size=24, color=YELLOW)341        not_converge_neg1.next_to(exists_text_25, DOWN, buff=0.2).shift(DOWN*0.35)342        self.play(Write(not_converge_neg1))343        self.wait(2)344        345        # 不淡出任何直线图像和文字346        self.play(347            # 保留所有元素：坐标轴、点、极限线、epsilon带、文字等348            even_points.animate.set_color(BLUE)349        )350 351        final_conclusion = Text("数列 ", font="SimSun", font_size=24)352        formula = MathTex(r"a_n = (-1)^n", font_size=24)353        final_text = Text(" 没有极限", font="SimSun", font_size=24)354        final_conclusion_group = VGroup(final_conclusion, formula, final_text).arrange(RIGHT, buff=0.1)355        final_conclusion_group.next_to(not_converge_neg1, DOWN, buff=0.2)  # 放在"数列不以-1为极限"正下方356        self.play(Write(final_conclusion_group))357        self.wait(2)358        359        # Show formal definition360        formal_def = MathTex(361            r"\exists \varepsilon > 0, \forall N, \exists n > N, |a_n - L| \geq \varepsilon",362            font_size=24363        )364        formal_def.next_to(final_conclusion_group, DOWN, buff=0.3)365        self.play(Write(formal_def))366        self.wait(3)