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| from manimlib.imports import * | ||
| import numpy as np | ||
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| def intro_text(): | ||
| return TextMobject("Cauchy") | ||
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| class intro(Scene): | ||
| def construct(self): | ||
| text=intro_text() | ||
| text1=TextMobject("Sequence") | ||
| text1.scale(1) | ||
| text.set_color_by_tex_to_color_map({"Cauchy": BLUE}) | ||
| text.scale(4) | ||
| text.move_to(1*UP+1*LEFT) | ||
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| self.play(Write(text)) | ||
| self.play(FadeIn(text1)) | ||
| self.wait(1) | ||
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| self.play(ApplyMethod(text.shift,1.5*UP),ApplyMethod(text1.shift,2.3*UP+3.2*RIGHT)) | ||
| self.wait(1) | ||
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| text2=TextMobject("what does","it","mean") | ||
| question=TextMobject("?") | ||
| text2.set_color_by_tex_to_color_map({"it":GREEN}) | ||
| text2.shift(0.5*LEFT) | ||
| text2.scale(1.5) | ||
| question.scale(4) | ||
| question.next_to(text2,RIGHT,buff=0.6) | ||
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| self.play(Write(text2)) | ||
| self.play(ShowCreation(question)) | ||
| self.wait(0.5) | ||
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| self.play(ApplyMethod(question.scale,-1)) | ||
| self.wait(1) | ||
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| text3=TextMobject("Here goes the formal definition..") | ||
| self.play(FadeOut(question)) | ||
| self.play(ReplacementTransform(text2,text3)) | ||
| self.wait(3) | ||
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| class definition(Scene): | ||
| def construct(self): | ||
| definition_text_1=TextMobject("A sequence, $(x_n)$ is said to be a" ,"Cauchy", "Sequence if ","for all", "$\\epsilon > 0$") | ||
| definition_text_2=TextMobject("$\\exists$ \\ K($\\epsilon$) such that ","for all m,n $\geq$ $ K(\\epsilon)$,","$|x_m-x_n|<\\epsilon$") | ||
| definition_text_1.shift(UP) | ||
| definition_text_1.set_color_by_tex_to_color_map({"Cauchy": YELLOW,"$\\epsilon > 0$": BLUE,"for all":RED}) | ||
| definition_text_2.set_color_by_tex_to_color_map({"$|x_m-x_n|<\\epsilon$": YELLOW," \\ K($\\epsilon$) ":BLUE,"for all m,n $\geq$ $ K(\\epsilon)$,":BLUE}) | ||
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| self.play(Write(definition_text_1)) | ||
| self.play(Write(definition_text_2)) | ||
| #self.play(Applymethod()) | ||
| self.wait(5.5) | ||
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| class to_graph(Scene): | ||
| def construct(self): | ||
| t1=TextMobject("Let us understand it", "graphically") | ||
| t1.scale(1.8) | ||
| t2=TextMobject("Consider the following graph of sequence") | ||
| t3=TextMobject("$X_n = (20 \\frac{(-1)^n}{n}) + 4$") | ||
| t3.set_color_by_tex_to_color_map({"$X_n = (20 \\frac{(-1)^n}{n}) + 4$":GREEN}) | ||
| t1.set_color_by_tex_to_color_map({"graphically": YELLOW}) | ||
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| self.play(Write(t1)) | ||
| self.wait(2) | ||
| self.play(ApplyMethod(t1.shift,2.3*UP)) | ||
| self.play(Write(t2)) | ||
| self.wait(1) | ||
| self.play(ReplacementTransform(t2,t3)) | ||
| self.wait(3) | ||
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| class graph(GraphScene): | ||
| CONFIG = { | ||
| "x_min": -2.5, | ||
| "x_max": 35, | ||
| "y_min": -10, | ||
| "y_max": 10, | ||
| "graph_origin": ORIGIN+4*LEFT, | ||
| "function_color": RED, | ||
| "axes_color": GREEN, | ||
| "x_axis_label": "$n$", | ||
| "y_axis_label": "$x_n$", | ||
| "exclude_zero_label": True, | ||
| "x_labeled_nums": range(0, 36, 5), | ||
| "y_labeled_nums": range(-10,11,2) | ||
| } | ||
| def construct(self): | ||
| t3=TextMobject("$X_n = (20 \\frac{(-1)^n}{n}) + 4$") | ||
| t3.scale(0.7) | ||
| t3.to_edge(UP+RIGHT) | ||
| t3.set_color(RED) | ||
| self.add(t3) | ||
| self.setup_axes(animate=True) | ||
| self.wait(1) | ||
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| mathfunc = lambda x: 20*((-1)**x)/x + 4 | ||
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| points = [ Dot(color = RED, radius = 0.03) for dot in range(1,34) ] | ||
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| x_each_unit = self.x_axis_width / (self.x_max - self.x_min) | ||
| y_each_unit = self.y_axis_height / (self.y_max - self.y_min) | ||
| [points[counter].shift(self.graph_origin + counter * (RIGHT * x_each_unit) + mathfunc(counter) * (UP * y_each_unit)) for counter in range(1, len(points))] | ||
| for i in range(1,len(points)): | ||
| self.add(points[i]) | ||
| self.wait(0.15) | ||
| self.wait(2) | ||
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| limit=4 | ||
| epsilon=2 | ||
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| limit_line=DashedLine(start=self.graph_origin,end=4.2*RIGHT,color=BLUE) | ||
| limit_line.shift(y_each_unit*limit*UP) | ||
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| self.play(Write(limit_line)) | ||
| self.wait(2) | ||
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| intro_arrow=TextMobject("Since" ,"$\\epsilon$"," value is","arbitary") | ||
| intro_arrow.shift(2.8*UP) | ||
| intro_arrow.set_color_by_tex_to_color_map({"arbitary":YELLOW,"$\\epsilon$":YELLOW}) | ||
| intro_arrow.scale(0.8) | ||
| self.play(Write(intro_arrow)) | ||
| self.wait(2) | ||
| arrow_text=TextMobject("Let epsilon $=$ 3") | ||
| arrow_text.scale(0.8) | ||
| arrow_text.shift(2.8*UP) | ||
| self.play(ReplacementTransform(intro_arrow,arrow_text)) | ||
| self.wait(1) | ||
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| epsilon_line=Line(start=(4-(20/7))*y_each_unit*UP,end=4*UP*y_each_unit) | ||
| epsilon_line.shift(self.graph_origin+x_each_unit*RIGHT*7) | ||
| self.play(Write(epsilon_line)) | ||
| self.wait(2) | ||
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| text=TextMobject("$\\epsilon=3$") | ||
| text.shift(self.graph_origin+RIGHT*5*x_each_unit+UP*3*y_each_unit) | ||
| text.scale(0.7) | ||
| self.play(Write(text),FadeOut(arrow_text)) | ||
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| arrow_text1=TextMobject("Now according to definition,","$\\exists K(\\epsilon)$") | ||
| arrow_text1.shift(2.8*UP+0.2*RIGHT) | ||
| arrow_text1.set_color_by_tex_to_color_map({"$\\exists K(\\epsilon)$":BLUE}) | ||
| arrow_text1.scale(0.8) | ||
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| self.play(Write(arrow_text1)) | ||
| self.wait(1) | ||
| point=Dot(color=BLUE,radius=0.06) | ||
| point.shift(self.graph_origin+RIGHT*7*x_each_unit) | ||
| self.add(point) | ||
| point_text=TextMobject("K($\\epsilon$)") | ||
| point_text.scale(0.7) | ||
| point_text.shift(self.graph_origin+RIGHT*7.2*x_each_unit+y_each_unit*1.4*DOWN) | ||
| self.play(FadeIn(point_text)) | ||
| self.wait(1.5) | ||
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| arrow_text2=TextMobject("such that" ,"$\\forall$ natural numbers") | ||
| arrow_text2.set_color_by_tex_to_color_map({"$\\forall$ natural numbers":YELLOW}) | ||
| arrow_text2a=TextMobject("m,n $\geq$ K($\\epsilon$)") | ||
| arrow_text2a.set_color(BLUE) | ||
| arrow_text2.scale(0.8) | ||
| arrow_text2a.scale(0.8) | ||
| arrow_text2.shift(2.8*UP+0.3*RIGHT) | ||
| arrow_text2a.shift(2.2*UP+0.3*RIGHT) | ||
| arrow_group=VGroup(arrow_text2,arrow_text2a) | ||
| self.play(ReplacementTransform(arrow_text1,arrow_group)) | ||
| self.wait(2) | ||
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| m_and_n_points=[Dot(color=WHITE, radius=0.05) for point in range(1,29)] | ||
| [m_and_n_points[i].shift(self.graph_origin+RIGHT*8*x_each_unit+RIGHT*i*x_each_unit) for i in range(0,28)] | ||
| for j in range(0,28): | ||
| self.add(m_and_n_points[j]) | ||
| self.wait(0.1) | ||
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| #here show all m,n by zoming out | ||
| self.wait(2) | ||
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| text1=TextMobject("(Consider any two points)") | ||
| text1.shift(2*DOWN+RIGHT) | ||
| self.play(Write(text1)) | ||
| self.wait(1) | ||
| self.play(FadeOut(text1)) | ||
| for k in range(28): | ||
| self.remove(m_and_n_points[k]) | ||
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| self.wait(1) | ||
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| pointm1=Dot(color=BLUE,radius=0.05) | ||
| pointm1.shift(self.graph_origin + 9*RIGHT*x_each_unit) | ||
| pointn1=Dot(color=BLUE,radius=0.05) | ||
| pointn1.shift(self.graph_origin + 11*RIGHT*x_each_unit) | ||
| self.play(Write(pointm1),Write(pointn1)) | ||
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| m=TextMobject("m") | ||
| m.shift(self.graph_origin+9*x_each_unit*RIGHT+y_each_unit*DOWN*0.7) | ||
| m.set_color(RED) | ||
| m.scale(0.4) | ||
| self.play(Write(m)) | ||
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| n=TextMobject("n") | ||
| n.shift(self.graph_origin+11*x_each_unit*RIGHT+y_each_unit*DOWN*0.7) | ||
| n.set_color(RED) | ||
| n.scale(0.4) | ||
| self.play(Write(n)) | ||
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| self.wait(1) | ||
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| self.play(ApplyMethod(pointm1.move_to,self.graph_origin + 9*RIGHT*x_each_unit + (4-(20/9))*UP*y_each_unit),ApplyMethod(pointn1.move_to,self.graph_origin + 11*RIGHT*x_each_unit + (4-(20/11))*UP*y_each_unit)) | ||
| self.wait(1) | ||
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| arrow_text3=TextMobject("$|X_m-X_n| < \\epsilon$") | ||
| arrow_text3.set_color_by_tex_to_color_map({"$|X_m-X_n| < \\epsilon$":YELLOW}) | ||
| arrow_text3.shift(2.8*UP+0.3*RIGHT) | ||
| self.play(ReplacementTransform(arrow_group,arrow_text3)) | ||
| self.wait(2) | ||
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| upline=Line(start=(4-20/9)*UP*y_each_unit,end=(4-20/11)*UP*y_each_unit) | ||
| upline.shift(self.graph_origin+9*RIGHT*x_each_unit) | ||
| leftline=Line(start=11*RIGHT*x_each_unit,end=9*RIGHT*x_each_unit) | ||
| leftline.shift(self.graph_origin+(4-20/11)*UP*y_each_unit) | ||
| self.play(Write(leftline)) | ||
| self.wait(0.4) | ||
| self.play(Write(upline)) | ||
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| self.wait(2) | ||
| self.play(ApplyMethod(epsilon_line.move_to,self.graph_origin+6*DOWN*y_each_unit+7*RIGHT*x_each_unit),ApplyMethod(upline.move_to,self.graph_origin+6*DOWN*y_each_unit+12*RIGHT*x_each_unit)) | ||
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| self.play(FadeOut(text),FadeOut(leftline),FadeOut(pointm1),FadeOut(pointn1),FadeOut(m),FadeOut(n)) | ||
| self.wait(1) | ||
| x1=TextMobject("$\\epsilon$") | ||
| x2=TextMobject("$|X_m-X_n|$") | ||
| x1.shift(self.graph_origin+7*RIGHT*x_each_unit+9*DOWN*y_each_unit) | ||
| x1.scale(0.8) | ||
| x2.shift(self.graph_origin+14*RIGHT*x_each_unit+9*DOWN*y_each_unit) | ||
| x2.scale(0.5) | ||
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| self.play(FadeIn(x1),FadeIn(x2)) | ||
| self.wait(1) | ||
| greater=TextMobject("$>$") | ||
| greater.shift(self.graph_origin+10*RIGHT*x_each_unit+6*DOWN*y_each_unit) | ||
| self.add(greater) | ||
| self.wait(1) | ||
| greater1=TextMobject("$>$") | ||
| greater1.scale(0.5) | ||
| greater1.shift(self.graph_origin+10*RIGHT*x_each_unit+9*DOWN*y_each_unit) | ||
| self.add(greater1) | ||
| self.wait(2.8) | ||
| self.play(FadeOut(x1),FadeOut(x2),FadeOut(greater1)) | ||
| self.play(FadeOut(greater),FadeOut(upline)) | ||
| self.wait(0.4) | ||
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| self.play(ApplyMethod(epsilon_line.move_to,self.graph_origin+(2.5)*UP*y_each_unit+7*RIGHT*x_each_unit)) | ||
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| self.wait(1.9) | ||
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| class conclusion(Scene): | ||
| def construct(self): | ||
| endformula=TextMobject("$|X_m-X_n|<\\epsilon$") | ||
| endformula.set_color_by_tex_to_color_map({"$|X_m-X_n|<\\epsilon$":YELLOW}) | ||
| endformula.shift(2.8*UP+0.3*RIGHT) | ||
| self.add(endformula) | ||
| self.wait(1) | ||
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| self.play(ApplyMethod(endformula.move_to,1.5*UP)) | ||
| self.wait(1.3) | ||
| endtext=TextMobject("Similarlly","the condition is true","$\\forall$ natural numbers","m,n$\geq$$K(\\epsilon)$") | ||
| endtext.set_color_by_tex_to_color_map({"it is true":YELLOW,"$\\forall$ natural numbers":BLUE}) | ||
| self.play(Write(endtext)) | ||
| self.wait(2.7) | ||
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| class conclusion_final(Scene): | ||
| def construct(self): | ||
| final1=TextMobject("Since the","equation","satisfies the","condition") | ||
| final1.set_color_by_tex_to_color_map({"equation":YELLOW,"condition":BLUE}) | ||
| final1.shift(UP) | ||
| final2=TextMobject("$|X_m-X_n|<\\epsilon$") | ||
| final2.set_color(RED) | ||
| final2.scale(1.5) | ||
| final3=TextMobject("$\\forall \\epsilon>0$"," and ","$\\forall$ m,n$\geq$$K(\\epsilon)$") | ||
| final3.set_color_by_tex_to_color_map({"$\\forall \\epsilon>0$":BLUE,"$\\forall$ m,n$\geq$$K(\\epsilon)$":BLUE}) | ||
| final3.shift(DOWN) | ||
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| self.play(Write(final1)) | ||
| self.wait(1) | ||
| self.play(FadeIn(final2)) | ||
| self.wait(0.6) | ||
| self.play(Write(final3)) | ||
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| self.wait(3.5) | ||
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| group1=VGroup(final1,final2,final3) | ||
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| t1=TextMobject("That is,","for any","small positive distance","$\\epsilon$") | ||
| self.wait(0.5) | ||
| t1.set_color_by_tex_to_color_map({"for any":YELLOW,"small positive distance":BLUE}) | ||
| self.play(ReplacementTransform(group1,t1)) | ||
| self.play(ApplyMethod(t1.move_to,1.5*UP)) | ||
| t2=TextMobject("a","finite number","of elements of the sequence are","less than") | ||
| t2.set_color_by_tex_to_color_map({"finite number":RED,"less than":YELLOW}) | ||
| t2.shift(0.5*UP) | ||
| t3=TextMobject("that","given distance","from each other and the elements","become") | ||
| t3.set_color_by_tex_to_color_map({"given distance":YELLOW,"become":BLUE}) | ||
| t3.shift(0.5*DOWN) | ||
| t4=TextMobject("arbitarily close","to each other","as the sequence progresses.") | ||
| t4.set_color_by_tex_to_color_map({"arbitarily close":BLUE,"as the sequence progresses.":RED}) | ||
| t4.shift(1.5*DOWN) | ||
| self.play(Write(t2)) | ||
| self.play(Write(t3)) | ||
| self.play(Write(t4)) | ||
| self.wait(3) | ||
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| group3=VGroup(t2,t3,t4) | ||
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| group4=VGroup(t1,t2,t3,t4) | ||
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| text1=TextMobject("Hence it is called") | ||
| text1.shift(UP) | ||
| text2=TextMobject("Cauchy","Sequence") | ||
| text2.scale(1.7) | ||
| text2.set_color_by_tex_to_color_map({"Cauchy":GREEN,"Sequence":RED}) | ||
| group2=VGroup(text1,text2) | ||
| self.play(ReplacementTransform(group4,group2)) | ||
| self.wait(4.5) | ||
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