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from manimlib.imports import *
import math
class uniformlyConvergent(Scene):
def construct(self):
introText1=TextMobject("Again consider the","above","example")
introText2=TextMobject("Let","$g(x)=\\frac { 1 }{ 1+{ x }^{ 2 } }$","and","x=0.5 $\in$(-1,1)")
introText3=TextMobject("Lets analyse..","!")
introText1.scale(0.8)
introText2.scale(0.7)
introText3.scale(0.9)
introText3.shift(DOWN)
introText1.set_color_by_tex_to_color_map({"above":YELLOW})
introText2.set_color_by_tex_to_color_map({"$g(x)=\\frac { 1 }{ 1+{ x }^{ 2 } }$":BLUE,"x=0.5 $\in$(-1,1)":YELLOW})
introText3.set_color_by_tex_to_color_map({"!":GREEN})
self.play(Write(introText1))
self.wait(0.5)
self.play(FadeOut(introText1))
self.play(Write(introText2))
self.play(FadeIn(introText3))
self.wait(2)
def gety(x,n):
ans=0
for i in range(0,n+1):
if(i%2==0):
ans+=(math.pow(x,2*i))
else:
ans-=(math.pow(x,2*i))
return ans
def makeSeries(x,points,x_each_unit,y_each_unit):
p=0
for point in points:
y=gety(x,p)
point.shift(ORIGIN+RIGHT*x_each_unit*p+UP*y_each_unit*y)
p+=1
def makeLines(x,numPoints,x_each_unit,y_each_unit):
lines=[0]*numPoints
for i in range(0,numPoints-1):
y=gety(x,i)
y_next=gety(x,i+1)
lines[i]=Line(start=ORIGIN+RIGHT*x_each_unit*i+UP*y_each_unit*y,end=ORIGIN+RIGHT*x_each_unit*(i+1)+UP*y_each_unit*y_next,color=RED)
return lines
class graphScene(GraphScene,MovingCameraScene):
CONFIG = {
"x_min": -6,
"x_max": 6,
"y_min": -5,
"y_max": 5,
"graph_origin": ORIGIN,
"function_color": RED,
"axes_color": GREEN,
"x_axis_label": "$k$",
"y_axis_label": "$f(\\frac{1}{2})_k$",
"exclude_zero_label": True,
"x_axis_width":7,
"y_axis_height":7
}
def setup(self):
GraphScene.setup(self)
MovingCameraScene.setup(self)
def construct(self):
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)
sequence=TextMobject("$1$ , $1-(0.5)^2$ , $1-(0.5)^2+(0.5)^4..$")
sequence.set_color(RED)
sequence.scale(0.35)
sequence.to_edge(UP+RIGHT)
formula=TextMobject("$f(x)_{ k }=\sum _{ i=0 }^{ k }{ (-1)^{ i }(x)^{ 2i } } $")
formula.set_color(PURPLE_C)
formula.scale(0.4)
formula.shift(5.3*RIGHT+3*UP)
fLine=Line(start=ORIGIN+x_each_unit*6*LEFT,end=ORIGIN+x_each_unit*6*RIGHT)
fLine.shift(ORIGIN+(4/5)*y_each_unit*UP)
fLineText=TextMobject("$g(0.5)=\\frac { 4 }{ 5 } $")
fLineText.set_color(RED)
fLineText.scale(0.3)
fLineText.shift(UP*1.2*y_each_unit+RIGHT*x_each_unit+4*LEFT)
points=[Dot(radius=0.03,color=BLUE) for i in range(0,6)]
makeSeries(0.5,points,x_each_unit,y_each_unit)
lines=makeLines(0.5,6,x_each_unit,y_each_unit)
self.add(sequence)
self.add(formula)
self.setup_axes(animate=True)
self.play(Write(fLine))
self.add(fLineText)
for p in points:
self.add(p)
for p in range(0,5):
self.play(Write(lines[p]))
self.wait(0.5)
self.camera_frame.save_state()
self.camera_frame.set_width(0.6)
self.play(self.camera_frame.move_to, points[0])
self.wait(0.4)
self.play(self.camera_frame.move_to, points[1])
self.wait(0.4)
self.play(self.camera_frame.move_to, points[2])
self.wait(0.3)
self.play(self.camera_frame.move_to, points[3])
self.wait(1)
self.play(self.camera_frame.move_to,ORIGIN)
self.camera_frame.set_width(14)
self.wait(1)
explanation1=TextMobject("Since the series","converges","to")
explanation1.set_color_by_tex_to_color_map({"converges":YELLOW})
explanation2=TextMobject("$\\frac {4}{5}$")
explanation2.set_color(BLUE)
explanation3=TextMobject("Hence","$\\forall \epsilon>0$,","$\exists k$","such that,")
explanation3.set_color_by_tex_to_color_map({"$\\forall \epsilon>0$":BLUE,"$\exists k$":YELLOW})
explanation4=TextMobject("$\left| { f\left( \\frac { 1 }{ 2 } \\right) }_{ k }-\\frac { 4 }{ 5 } \\right| <$","$\epsilon$")
explanation4.set_color_by_tex_to_color_map({"$\epsilon$":RED})
explanation1.scale(0.5)
explanation3.scale(0.5)
explanation1.shift(1.8*DOWN+3.5*RIGHT)
explanation2.shift(2.4*DOWN+3.5*RIGHT)
explanation3.shift(1.8*DOWN+3.5*RIGHT)
explanation4.shift(2.4*DOWN+3.5*RIGHT)
self.play(Write(explanation1))
self.play(FadeIn(explanation2))
self.wait(1)
self.play(FadeOut(explanation1),FadeOut(explanation2))
self.play(Write(explanation3))
self.play(Write(explanation4))
self.wait(2)
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