交错级数收敛性

1# -*- coding: utf-8 -*-2from manim import *3import numpy as np4 5class AlternatingSeriesConvergence(Scene):6    def construct(self):7        # 设置默认字体8        Text.set_default(font="SimSun")9        10        # 标题11        title = Text("交错级数收敛性").scale(0.8)12        title.to_edge(UP, buff=0.5)13        self.play(Write(title))14        self.wait(1)15 16        # 创建交错级数的各项17        terms = VGroup()18        n_visible_terms = 6  # 减回到6项19        for i in range(n_visible_terms):20            if i == 0:21                term = MathTex("1")22            else:23                sign = "-" if i % 2 == 1 else "+"24                term = MathTex(f"{sign}\\frac{{1}}{{{i+1}}}")25            terms.add(term)26        terms.add(MathTex("\\cdots"))27        terms.arrange(RIGHT, buff=0.2)28        29        # 创建完整的级数表达式30        series = VGroup(31            MathTex("\\sum_{n=1}^{\\infty} (-1)^{n+1}\\frac{1}{n} ="),32            terms33        ).arrange(RIGHT, buff=0.3).scale(0.8)34        series.move_to(ORIGIN + UP * 2)  # 居中显示在上方35 36        # 添加收敛性说明（移到右上方）37        explanation = VGroup(38            Text("• 通项绝对值单调递减", font="SimSun"),39            Text("• 通项趋于零", font="SimSun"),40            Text("• 级数收敛于ln(2)", font="SimSun")41        ).arrange(DOWN, aligned_edge=LEFT).scale(0.5)42        explanation.to_corner(UR, buff=0.5)  # 移到右上角43        44        # 添加误差估计（跟随说明文字）45        error_bound = MathTex(46            r"|R_n| \leq \frac{1}{n+1}",47            color=BLUE48        ).scale(0.7)49        error_bound.next_to(explanation, DOWN, buff=0.5)50        51        error_text = Text(52            "误差界估计",53            font="SimSun"54        ).scale(0.5).next_to(error_bound, UP, buff=0.2)55 56        self.play(Write(series))57        self.wait(1)58 59        # 创建坐标系（居中显示）60        axes = Axes(61            x_range=[0, 20, 2],62            y_range=[-0.5, 1.5, 0.5],63            axis_config={64                "include_tip": True,65                "include_numbers": False66            },67            x_length=10,  # 增加宽度68            y_length=6    # 增加高度69        ).move_to(ORIGIN + DOWN * 0.5)  # 居中显示，稍微下移70 71        # 添加分数刻度标签72        y_labels = VGroup()73        y_values = [-0.5, 0, 0.5, 1, 1.5]74        for y in y_values:75            if y == 0:76                label = MathTex("0")77            elif y == 1:78                label = MathTex("1")79            else:80                num, den = float(y).as_integer_ratio()81                label = MathTex(f"\\frac{{{num}}}{{{den}}}")82            label.scale(0.5)83            label.next_to(axes.c2p(0, y), LEFT, buff=0.1)84            y_labels.add(label)85 86        # 添加x轴刻度标签（使用整数）87        x_labels = VGroup()88        for x in range(0, 21, 2):89            if x > 0:  # 跳过原点90                label = MathTex(f"{x}").scale(0.5)91                label.next_to(axes.c2p(x, 0), DOWN, buff=0.1)92                x_labels.add(label)93 94        # 添加坐标轴标签95        x_label = Text("n", font="SimSun").scale(0.6).next_to(axes.x_axis, RIGHT)96        y_label = Text("部分和", font="SimSun").scale(0.6).next_to(axes.y_axis, UP)97        98        self.play(99            Create(axes),100            Write(x_label),101            Write(y_label),102            Write(x_labels),103            Write(y_labels)104        )105 106        # 计算部分和107        def get_partial_sum(n):108            return sum((-1)**(k+1)/k for k in range(1, n+1))109 110        # 创建高亮框111        highlight_box = SurroundingRectangle(terms[0], color=YELLOW)112        self.play(Create(highlight_box))113 114        # 创建部分和点和折线115        dots = VGroup()116        n_terms = 20117        partial_sums = [get_partial_sum(n) for n in range(1, n_terms+1)]118        119        # 创建点120        prev_dot = None121        for i, sum_value in enumerate(partial_sums):122            dot = Dot(axes.c2p(i+1, sum_value), color=RED)123            dots.add(dot)124            125            if prev_dot:126                line = Line(prev_dot.get_center(), dot.get_center(), color=YELLOW)127                self.play(128                    Create(dot),129                    Create(line),130                    run_time=0.3131                )132            else:133                self.play(Create(dot), run_time=0.3)134            prev_dot = dot135 136            # 更新高亮框位置137            if i < len(terms) - 1:138                new_box = SurroundingRectangle(terms[:i+1], color=YELLOW)139                self.play(140                    Transform(highlight_box, new_box),141                    run_time=0.3142                )143 144        # 显示极限值145        limit_value = np.log(2)146        limit_line = DashedLine(147            axes.c2p(0, limit_value),148            axes.c2p(20, limit_value),149            color=GREEN150        )151        limit_label = MathTex(r"\ln(2)", color=GREEN).scale(0.8)152        limit_label.next_to(limit_line.get_end(), RIGHT)153 154        self.play(155            Create(limit_line),156            Write(limit_label)157        )158 159        # 添加收敛性说明160        explanation = VGroup(161            Text("• 通项绝对值单调递减", font="SimSun"),162            Text("• 通项趋于零", font="SimSun"),163            Text("• 级数收敛于ln(2)", font="SimSun")164        ).arrange(DOWN, aligned_edge=LEFT).scale(0.5)165        explanation.to_corner(UR, buff=0.5)  # 移到右上角166        167        self.play(Write(explanation))168        169        # 显示振荡收敛的特点170        oscillation = Text(171            "交错级数的部分和在极限值附近振荡收敛",172            font="SimSun"173        ).scale(0.5)174        oscillation.to_edge(DOWN, buff=0.5)175        176        self.play(Write(oscillation))177        self.wait(2)178 179        # 添加误差估计180        error_bound = MathTex(181            r"|R_n| \leq \frac{1}{n+1}",182            color=BLUE183        ).scale(0.7)184        error_bound.next_to(explanation, DOWN, buff=0.5)185        186        error_text = Text(187            "误差界估计",188            font="SimSun"189        ).scale(0.5).next_to(error_bound, UP, buff=0.2)190        191        self.play(192            Write(error_text),193            Write(error_bound)194        )195        self.wait(2)