Create a Program to Compute Numerical Methods in Java Assignment Solution.

from sympy import *

from sympy.abc import x

from sympy import Symbol

import numpy as np

from sympy.plotting import plot

import matplotlib.pyplot as plt

def a(n):

return (x ** (3 * n + 1)) / (3 * n + 1)

def PartA():

print("*** PART A ****")

# Calculate the ratio test

n = symbols('n', integer = True, positive = True)

# Print simplified version of a(n+1)/a(n)

simplified = simplify(a(n+1)/a(n))

print("Simplified: ")

print(simplified)

print()

# Now calculate the limit

print("Limit: ")

print(limit(simplified, n, float('inf')))

print()

def PartB():

print("*** PART B ****")

# Calculate the ratio test

n = symbols('n', integer=True, positive=True)

# Print simplified version of a(n+1)/a(n)

simplified = simplify(a(n + 1) / a(n))

# Now calculate the limit

lim = limit(simplified, n, float('inf'))

# Calculate the intervals for which x**3 < 1

interval = reduce_inequalities(lim<1)

print("Interval of convergence:")

print(interval)

print()

""" Each endpoint is 1 value far from x = 0 (-1, 1)"""

def PartC():

print("*** PART C ****")

# Define the function of convergence

eq = 1/3 *log(x+1) - 1/6 *log(x**2-x+1) + 1/sqrt(3) *atan((2*x-1)/sqrt(3)) + pi*sqrt(3)/18

# Interval of convergence is [-1,1)

# Define x values

xvals = np.linspace(-1, 1-0.01, 100)

yvals = lambdify(x,eq)(xvals)

# Now plot

plt.plot(xvals, yvals, label = 'f(x)')

# Compute sq

s1 = a(0) + a(1)

s3 = a(0) + a(1) + a(2) + a(3)

s5 = a(0) + a(1) + a(2) + a(3) + a(4) + a(5)

y_s1 = lambdify(x, s1)(xvals)

y_s3 = lambdify(x, s3)(xvals)

y_s5 = lambdify(x, s5)(xvals)

plt.plot(xvals, y_s1, label = 's1')

plt.plot(xvals, y_s3, label='s3')

plt.plot(xvals, y_s5, label='s5')

plt.grid(True)

plt.legend()

plt.show()

print()

def PartD():

print("*** PART D ****")

# Define t

t = symbols('t')

#Define n

n = symbols('n', integer=True, positive=True)

integral1 = integrate(1/(1+t**3), (t, 0, x))

integral2 = integrate((-1)**n *t**(3*n), (t, 0, x))

print(integral1)

print()

print(integral2)

xvals = np.linspace(-1, 1 - 0.01, 100)

yvals1 = lambdify(x, integral1)(xvals)

yvals2 = lambdify(x, integral2)(xvals)

"""

From Part C), we found that the series converges to the following function:

1/3 *log(x+1) - 1/6 *log(x**2-x+1) + 1/sqrt(3) *atan((2*x-1)/sqrt(3)) + pi*sqrt(3)/18

And we see that the value of the first integral is

log(x + 1)/3 - log(x**2 - x + 1)/6 + sqrt(3)*atan(2*sqrt(3)*x/3 - sqrt(3)/3)/3 + sqrt(3)*pi/18

Which are the same functions.

"""

if __name__ == '__main__':

PartA()

PartB()

PartC()

PartD()

from sympy import *

from sympy.abc import x

from sympy import Symbol

import numpy as np

from sympy.plotting import plot

import matplotlib.pyplot as plt

def a(n):

return x**n *factorial(2*n)/(factorial(n)**2)

def PartA():

print("*** Part A ***")

# Calculate the ratio test

n = symbols('n', integer=True, positive=True)

# Print simplified version of a(n+1)/a(n)

simplified = simplify(a(n + 1) / a(n))

print("Simplified: ")

print(simplified)

print()

# Now calculate the limit

print("Limit: ")

print(limit(simplified, n, float('inf')))

print()

def PartB():

print("*** Part B ***")

# Calculate the ratio test

n = symbols('n', integer=True, positive=True)

# Print simplified version of a(n+1)/a(n)

simplified = simplify(a(n + 1) / a(n))

# Now calculate the limit

lim = limit(simplified, n, float('inf'))

# Calculate the intervals for which x**3 < 1

interval = reduce_inequalities(lim < 1)

print("Interval of convergence:")

print(interval)

print()

def PartC():

print("*** Part C ***")

# Define the function of convergence

eq = 1/sqrt(1-4*x)

# Interval of convergence is [-1,1)

# Define x values

xvals = np.linspace(-1/4, 1/4 - 0.01, 100)

yvals = lambdify(x, eq)(xvals)

# Now plot

plt.plot(xvals, yvals, label='f(x)')

# Compute sq

s5 = a(0)

for i in range(1, 6):

s5 += a(i)

s10 = a(0)

for i in range(1, 11):

s10 += a(i)

s15 = a(0)

for i in range(1, 16):

s15 += a(i)

y_s5 = lambdify(x, s5)(xvals)

y_s10 = lambdify(x, s10)(xvals)

y_s15 = lambdify(x, s15)(xvals)

plt.plot(xvals, y_s5, label='s5')

plt.plot(xvals, y_s10, label='s10')

plt.plot(xvals, y_s15, label='s15')

plt.grid(True)

plt.legend()

plt.show()

if __name__ == '__main__':

PartA()

PartB()

PartC()

from sympy import *

from sympy import Symbol

import numpy as np

from sympy.plotting import plot

import matplotlib.pyplot as plt

# Define x as positive

x = symbols('x', positive = True)

def a(n):

return x**(-log(n))

def PartA():

print("*** Part A ***")

# Calculate the ratio test

n = symbols('n', integer=True, positive=True)

# Print simplified version of a(n+1)/a(n)

simplified = simplify(a(n + 1) / a(n))

print("Simplified: ")

print(simplified)

print()

# Now calculate the limit

print("Limit: ")

print(limit(simplified, n, float('inf')))

print()

"""

We see that the result of the limit is 1. This means that the Ratio Test is inconclusive

"""

def PartB():

print("*** Part B ***")

# Calculate the ratio test

n = symbols('n', integer=True, positive=True)

# Define bn

xx = .9

bn = abs(xx**(-log(n)))

# Now, find the limit

print("Limit: ")

print(limit(bn, n, float('inf')))

print()

"""

In this case, the result is that the limit goes to infinity, which means that

the series diverges

"""

def PartC():

print("*** Part C ***")

# Calculate the ratio test

n = symbols('n', integer=True, positive=True)

# Define bn

xx = 1.1

cn = abs(xx ** (-log(n)))

# Now, find the limit

print("Limit: ")

print(limit(cn, n, float('inf')))

print()

"""

In this case, the result is that the limit goes to zero. It means that for x > 1 the

series converges.

"""

def PartD():

print("*** Part D ***")

print("There is no interval of convergence")

def PartE():

print("*** Part E ***")

"""

Using properties of logarithms, we have that, for x = e:

x^(-log(n)) = exp(-log(n)) = 1/exp(log(n)) = 1/n

Then the series is:

(-1)^(n+1) *(1/n) = -(-1)^n *(1/n)

(-1)^n (1/n) is a known series and it is known that its sum is log(2)

"""

xx = exp(1) # x = e

# Calulate the sum

n = symbols('n', integer=True, positive=True)

an = (-1)**(n+1) *xx**(-log(n))

# Define the values of n from 1 to a big number

nvals = np.linspace(1, 1E6, int(1E6))

sum_result = sum(lambdify(n, an)(nvals))

# Print sum

print("The calculated sum of the series is: {:.5f}".format(sum_result))

print("The expected sum of the series is: {:.5f}".format(np.log(2))) # print the value of log(2)

if __name__ == '__main__':

PartA()

PartB()

PartC()

PartD()

PartE()