Review for the Common Exam: MATH 152 Exam 3 b Review Problems 1-3
Review of sequences and finding the sum of a series
Generating and Plotting Partials Sums of a Series in Python
Author: David Manuel
In this video, we use Python to compute and plot the partial sums of a series in order to estimate the sum of the corresponding infinite series. Then we use Python to calculate the exact value of the series.
Transcript 26
Transcript 26
Problem: Given the series
\[\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n^2}\]
- Find and plot the first 20 partial sums.
- Find the exact sum of the series.
Python Code:
Given the series sum from 1 to infinity of (-1)^(n+1)/n^2
a) Find and plot the first 20 partial sums
b) Find the exact sum of the series
from sympy import *
import numpy as np
n=symbols('n',positive=True,integer=True)
a=(-1)**(n+1)/n**2
n1to20=range(1,21)
a1to20=[a.subs(n,i).evalf() for i in n1to20]
print(a1to20)
s1to20=np.cumsum(a1to20)
print(s1to20)
matplotlib notebook
import matplotlib.pyplot as plt
plt.plot(n1to20,s1to20,'o')
from mpmath import nsum,inf
S=nsum(lambda n:(-1)**(n+1)/n**2,[1,inf])
print('The sum of the series is',S)
Open Python Notebook File
Given the series sum from 1 to infinity of (-1)^(n+1)/n^2
a) Find and plot the first 20 partial sums
b) Find the exact sum of the series
from sympy import *
import numpy as np
n=symbols('n',positive=True,integer=True)
a=(-1)**(n+1)/n**2
n1to20=range(1,21)
a1to20=[a.subs(n,i).evalf() for i in n1to20]
print(a1to20)
s1to20=np.cumsum(a1to20)
print(s1to20)
matplotlib notebook
import matplotlib.pyplot as plt
plt.plot(n1to20,s1to20,'o')
from mpmath import nsum,inf
S=nsum(lambda n:(-1)**(n+1)/n**2,[1,inf])
print('The sum of the series is',S)
Open Python Notebook File
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An introduction to installing and getting started with Python
List Comprehension and Patterns in Higher Derivatives in Python
Using Python's list comprehension tool to find several higher order derivatives of a function at once
Parametric Plots and Tangents at a Given t-value in Python
Using Python to find the tangent line to a parametric equation and plot the two graphs
Implicit Plots and Implicit Differentiation in Python
Using Python to plot an implicit curve and find a tangent line using implicit differentiation
Definite and Indefinite Integrals in Python
Solving definite and indefinite integrals in Python
Finding Absolute (Global) Extrema in Python
Using Python to find the absolute maximum and minimum of a function on a closed interval
Finding and Graphing Tangent Lines to a Curve in Python
Using Python to find the equation of the tangent line to a curve and graphing the result
Finding Derivatives in Python
Finding derivatives in Python and solving for the rate of change of a force
Finding Horizontal Tangent Lines in Python
Finding when the tangent line to a function is horizontal in Python
Finding Intervals of Increase/Decrease and Concavity in Python
Using Python to find the intervals where a function is increasing or decreasing
Approximating Definite Integrals with Riemann Sums in Python
Using Python to approximate a definite integral using left endpoint Riemann sums
Word Problems and the Algorithmic Process for Multistep Problem Solving in Python
Solving a multistep word problem in Python and graphing the resulting function
Tangents to Parametrized Curves at a Given Point in Python
Using Python to plot a parametrized curve and its tangent line at a point
Volumes of Revolution in Python
Finding the volume of a solid of revolution in Python
Areas Between Curves in Python
Using Python to identify the region between curves and then evaluate the area of the region
Calculating Work in Python: Hooke’s Law
Using Python to calculate the amount of work done in moving a spring using Hooke's Law
Calculating Work in Python: Pumping Liquid from a Tank
Using Python to calculate the work done in pumping the liquid from a tank whose ends are semicircles
For Loops and Recursive Sequences in Python
Using for loops in Python to generate a recursively defined sequence
Radius and Interval of Convergence in Python
Using Python to find the radius and interval of convergence of a power series with the Ratio Test
Derivatives of Exponential Functions
Explaining the formulas for the derivatives of exponential functions
Review for the Common Exam: MATH 151 Exam 1 Review Problems 14-25
Review of limits, continuity, and the Intermediate Value Theorem
Review for the Common Exam: MATH 151 Exam 1 Review Problems 26-35
Review of the limit definition of a derivative and calculating the derivative
Review for the Common Exam: MATH 151 Exam 3 Review Problems 8-15
Review of limits and derivatives of inverse trigonometric functions