CoCalc -- 2023-04-10-file-1.ipynb

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ubuntu2004

Kernel: SageMath 9.8

In [26]:

n=var('n')
def a(n):
    return x^(n+1)/((n+1)*5^(n+1))

In [27]:

r=abs(a(n+1)/a(n))
rho=limit(r,n=infinity)
show(rho)

#this is the ratio test basically

Out[27]:

$\displaystyle \frac{1}{5} \, {\left| x \right|}$

In [28]:

solve(rho<1,x)

Out[28]:

[[-5 < x, x < 0], [x == 0], [0 < x, x < 5]]

In [29]:

forget()
assume(x<5)
assume(x>-5)
S=sum(a(n),n,0,infinity)
show(S)
#if final project asks to draw a conclusion, thne test the endpoints
# ex: test the endpoints 1 and -1

Out[29]:

$\displaystyle -\log\left(-\frac{1}{5} \, x + 1\right)$

In [30]:

def f(x):
    return -log(-(1/5)*x+1)

In [32]:

f(-5)

Out[32]:

-log(2)

In [33]:

f(5)

Out[33]:

+Infinity

Example 2: (n(n-1)*x^(n-2))

In [34]:

n=var('n')
def a(n):
    return (n*(n-1)*x^(n-2))

In [35]:

r=abs(a(n+1)/a(n))
rho=limit(r,n=infinity)
show(rho)

Out[35]:

$\displaystyle {\left| x \right|}$

In [36]:

solve(rho<1,x)

Out[36]:

[[-1 < x, x < 1]]

In [37]:

forget()
assume(x<1)
assume(x>-1)
S=sum(a(n),n,0,infinity)
show(S)

Out[37]:

$\displaystyle -\frac{2}{x^{3} - 3 \, x^{2} + 3 \, x - 1}$

In [39]:

def f(x):
    return -((2)/((x^3)-(3*x^2)+3*x-1))

In [40]:

f(-1)

Out[40]:

1/4

In [46]:

f(1)
#wont work bc division by 0 so do this endpoint by hand

Out[46]:

---------------------------------------------------------------------------
ZeroDivisionError                         Traceback (most recent call last)
Cell In [46], line 1
----> 1 limit(f(Integer(1)),x=Integer(1))
Cell In [39], line 2, in f(x)
      1 def f(x):
----> 2     return -((Integer(2))/((x**Integer(3))-(Integer(3)*x**Integer(2))+Integer(3)*x-Integer(1)))
File /ext/sage/9.8/src/sage/rings/integer.pyx:2019, in sage.rings.integer.Integer.__truediv__()
   2017 if type(left) is type(right):
   2018     if mpz_sgn((<Integer>right).value) == 0:
-> 2019         raise ZeroDivisionError("rational division by zero")
   2020     x = <Rational> Rational.__new__(Rational)
   2021     mpq_div_zz(x.value, (<Integer>left).value, (<Integer>right).value)
ZeroDivisionError: rational division by zero

In [0]:

Approximate Taylor Polynomial

approximate e^x with 1,2, and 3 degree

In [49]:

n=var('n')
@interact
def _(m=input_box(2,label='Degree',width=10)):
    print("App1=",N(sum(1/factorial(n),n,0,m),digits=5))
    
#ex: if sin(x) you would use sin(2) for the approximation

Out[49]:

Interactive function <function _ at 0x7f500ef4bf60> with 1 widget
  m: EvalText(value='2', description='Degree…

In [50]:

n=var('n')
@interact
def _(j=input_box(3,label='Degree',width=10)):
    print("App2=",N(sum(1/factorial(n),n,0,j),digits=5))

Out[50]:

Interactive function <function _ at 0x7f500eec45e0> with 1 widget
  j: EvalText(value='3', description='Degree…

In [51]:

n=var('n')
@interact
def _(k=input_box(4,label='Degree',width=10)):
    print("App3=",N(sum(1/factorial(n),n,0,k),digits=5))

Out[51]:

Interactive function <function _ at 0x7f500ef485e0> with 1 widget
  k: EvalText(value='4', description='Degree…

In [59]:

n=var('n')
plot(sum(x^n/factorial(n),n,0,2),xmin=-10,xmax=10,color='blue',legend_label='degree2')+plot(sum(x^n/factorial(n),n,0,3),xmin=-10,xmax=10,color='red',legend_label='degree3')+plot(sum(x^n/factorial(n),n,0,4),xmin=-10,xmax=10,color='green',legend_label='degree4')+plot(e^x,xmin=-10,xmax=10,ymax=1000,color='black',legend_label='$e^x$')

Out[59]:

In [0]:

Product

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