WIR 20B M152 V75
Finding the interval and radius of convergence for a power series
tags:
Alternating Series Test Comparison Test Interval of Convergence Power Series Power Series Convergence Python Radius of Convergence Ratio TestRadius and Interval of Convergence in Python
Author: David Manuel
In this video, we demonstrate how to use Python to find the radius and interval of convergence of a power series. We will use the Ratio Test when determining the radius of convergence. When testing the end points of the interval of convergence, we use the Comparison Test and the Alternating Series Test.
Transcript 28
Transcript 28
Problem: Find the radius and interval of convergence of
\[\sum_{n=0}^\infty \frac{(-1)^n x^n}{ (n+2) 2^n}\]
Python Code:
Find the radius and interval of convergence of the sum from 0 to infinity of (-1)^n * x^n/( (n+2) * 2^n)
from sympy import *
n=symbols('n',positive=True,integer=True)
x=symbols('x')
a=(-1)**n*x**n/((n+2)*2**n)
RatioTest=abs(a.subs(n,n+1)/a)
print(RatioTest.simplify())
L=limit(RatioTest.simplify(),n,oo)
print('The limit of the Ratio Test is',L)
# likely obvious x is between -2 and 2
reduce_inequalities(L<1)
# Left endpoint
aLeft=a.subs(x,-2)
print(aLeft.simplify(),'This series diverges by limit comparison with the series 1/n.')
# Right endpoint
aRight=a.subs(x,2)
print(aRight.simplify(),'This series converges by the Alternating Series Test.')
Open Python Notebook File
Find the radius and interval of convergence of the sum from 0 to infinity of (-1)^n * x^n/( (n+2) * 2^n)
from sympy import *
n=symbols('n',positive=True,integer=True)
x=symbols('x')
a=(-1)**n*x**n/((n+2)*2**n)
RatioTest=abs(a.subs(n,n+1)/a)
print(RatioTest.simplify())
L=limit(RatioTest.simplify(),n,oo)
print('The limit of the Ratio Test is',L)
# likely obvious x is between -2 and 2
reduce_inequalities(L<1)
# Left endpoint
aLeft=a.subs(x,-2)
print(aLeft.simplify(),'This series diverges by limit comparison with the series 1/n.')
# Right endpoint
aRight=a.subs(x,2)
print(aRight.simplify(),'This series converges by the Alternating Series Test.')
Open Python Notebook File
See more videos from this section
Related Videos
Review for the Common Exam: MATH 152 Exam 3 c Review Problem 9
Review of finding the radius and interval of convergence for a power series
WIR 20B M152 V73
Finding the interval and radius of convergence for a power series
Review for the Common Exam: MATH 152 Exam 3 c Review Problem 9
Review of finding the radius and interval of convergence for a power series
WIR 20B M152 V74
Finding the interval and radius of convergence for a power series
Radius and Interval of Convergence for Power Series: MATH 152 11.8 Problem 1
Determining the radius and interval of convergence for a power series
Radius and Interval of Convergence for Power Series: MATH 152 11.8 Problem 1
Determining the radius and interval of convergence for a power series
Power Series: MATH 172 Problems 1-3
Covering Radius and Interval of Convergence of Power Series
Power Series as Functions: MATH 172 Problems 4 & 5
Examples about the derivative and integral of a power series
MATH 308 WIR22A V70
Determining the radius of convergence of a power series
Review for the Common Exam: MATH 152 Exam 3 b Review Problems 4 & 5
Review of using tests to determine if a series is absolutely convergent, convergent, and divergent
Review for the Common Exam: MATH 152 Exam 3 c Review Problems 10-13
Review of power series properties and writing functions as power series
Review for the Common Exam: MATH 152 Exam 3 c Review Problems 4 & 5
Review of using tests to determine if a series is absolutely convergent, convergent, and divergent
Representations of Functions as Power Series: MATH 152 11.9 Problems 2 & 3
Finding power series representations for functions and the radius and interval of convergence of the power series
Review for the Common Exam: MATH 152 Exam 3 b Review Problems 4 & 5
Review of using tests to determine if a series is absolutely convergent, convergent, and divergent
Review for the Common Exam: MATH 152 Exam 3 c Review Problems 10-13
Review of power series properties and writing functions as power series
Review for the Common Exam: MATH 152 Exam 3 c Review Problems 4 & 5
Review of using tests to determine if a series is absolutely convergent, convergent, and divergent
Alternating Series and Ratio Tests: MATH 172 Problems 8-10
Proving the Alternating Series and Ratio Test and covers absolute convergence
Representations of Functions as Power Series: MATH 152 11.9 Problems 2 & 3
Finding power series representations for functions and the radius and interval of convergence of the power series
MATH 308 WIR22A V71
Finding a power series solution to a differential equation
MATH 308 WIR22A V72
Finding a power series solution to a differential equation
Taylor and Maclaurin Series: MATH 152 11.10 Problems 4-10
Finding Taylor and Maclaurin Series for functions
WIR 20B M152 V68
Determining if a series is absolutely convergent, conditionally convergent, or divergent.
WIR 20B M152 V76
Finding a power series representation for a function
WIR 20B M152 V77
Finding a power series representation for a function
Taylor and Maclaurin Series: MATH 152 11.10 Problems 4-10
Finding Taylor and Maclaurin Series for functions
Math 152: Alternating Series Exercise 5
Determining if an alternating series converges or diverges
Plotting a Function and Partial Sums of its Power Series in Python
Video 29 in a series of Python instructional videos
Review for the Common Exam: MATH 152 Exam 3 c Review Problems 7 & 8
Review of estimating the sum of a series and using the Alternating Series Estimation Theorem
WIR 20B M152 V69
Determining if a series is absolutely convergent, conditionally convergent, or divergent.
WIR 20B M152 V70
Determining if a series is absolutely convergent, conditionally convergent, or divergent.
WIR 20B M152 V71
Determining if a series is absolutely convergent, conditionally convergent, or divergent.
WIR 20B M152 V72
Determining if a series is absolutely convergent, conditionally convergent, or divergent.
Absolute Convergence and the Ratio Test: MATH 152 11.6 Problems 3(d-f) & 4
Using the Ratio Test to determine if series are absolutely convergent
WIR 20B M152 V78
Evaluating an indefinite integral using a power series
Review for the Common Exam: MATH 152 Exam 3 c Review Problems 7 & 8
Review of estimating the sum of a series and using the Alternating Series Estimation Theorem
Integral and Comparison Tests: MATH 172 Problems 4-7
Proving the Integral, Comparison, and Limit Comparison Tests
The Comparison Test for Series
Explaining the Comparison Test to determine if series are convergent or divergent
Absolute Convergence and the Ratio Test: MATH 152 11.6 Problems 3(d-f) & 4
Using the Ratio Test to determine if series are absolutely convergent
MATH 308 WIR22A V73
Finding a power series solution to an initial value problem
MATH 308 WIR22A V74
Finding the value of the second and third derivative of a solution at the given point
Scalar-Valued IVP
Showing how the exponential function is the solution to an initial value problem
System of Differential Equations Video 1
Using power series to define a matrix exponential function to use in solving differential equations with matrices
Systems of Differential Equations Exercise 4
Solving a system of first-order linear differential equations using a matrix exponential function
Math 152: Series Exercise 4
Finding properties of a series given a formula for the partial sums of the series
Math 152: Series Exercise 12
Solving for a variable to make a geometric series converge
The Comparison Test for Series
Explaining the Comparison Test to determine if series are convergent or divergent
Math 152: Comparison Tests Exercise 1
Using the Comparison Test to draw a conclusion about whether a series converges or diverges
Math 152: Comparison Tests Exercise 2
Using the Comparison Test to draw a conclusion about whether a series converges or diverges
Math 152: Comparison Tests Exercise 3
Using the Comparison Test to determine if a series converges or diverges
Math 152: Comparison Tests Exercise 4
Using the Comparison Test to determine if a series converges or diverges
Math 152: Comparison Tests Exercise 5
Using the Comparison Test to determine if a series converges or diverges
Math 152: Comparison Tests Exercise 6
Using the Comparison Test to determine if a series converges or diverges
Math 152: Comparison Tests Exercise 7
Using the Comparison Test to determine if a series converges or diverges
WIR 20B M152 V55
Using the comparison test to determine if a series converges or diverges
WIR 20B M152 V56
Using the comparison test to determine if a series converges or diverges
WIR 20B M152 V57
Using the comparison test to determine if a series converges or diverges
WIR 20B M152 V58
Using the comparison test to determine if a series converges or diverges
WIR 20B M152 V62
Using the Alternating Series Test to determine if a series converges or diverges
WIR 20B M152 V63
Using the Alternating Series Test to determine if a series converges or diverges
WIR 20B M152 V64
Using the Alternating Series Test to determine if a series converges or diverges
WIR 20B M152 V65
Determining if an Alternating Series converges or diverges
Math 152: Alternating Series Exercise 1
Determining if an alternating series converges or diverges
Math 152: Alternating Series Exercise 3
Determining if an alternating series converges or diverges
Math 152: Alternating Series Exercise 6
Proving an alternating series converges and estimating its sum and the error
Math 152: Alternating Series Exercise 7
Approximating an alternating series with a partial sum to within a certain accuracy
Solving Equations Symbolically in Python
Solving equations symbolically in Python and interpreting the results
Using Variables in Python
Defining variables in Python and using the variables in equations
Introduction and Installation of Python
An introduction to installing and getting started with Python
Numerical Calculations in Python
Performing basic numerical calculations in Python
Creating and Manipulating Symbolic Expressions in Python
Creating symbolic expression in Python and then factoring, expanding, and simplifying the expressions
Plotting Expressions and Solving Equations Numerically in Python
Plotting an expression in Python and finding a numerical approximation for the solution
Plotting Multiple Expressions in One Graph in Python
Graphing a piecewise function using Python
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
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
Simplifying Difference Quotients in Python
Using Python to simplify a difference quotient and find a derivative using the limit definition
List Comprehension and Numerical, Graphical, and Symbolic Limits in Python
Using Python to numerically estimate a limit, graphically estimate a limit, and find the exact limit
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
Approximating Definite Integrals with Riemann Sums in Python
Using Python to approximate a definite integral using left endpoint Riemann sums
Tangents to Parametrized Curves at a Given Point in Python
Using Python to plot a parametrized curve and its tangent line at a point
Implicit Plots and Implicit Differentiation in Python
Using Python to plot an implicit curve and find a tangent line using implicit differentiation
Word Problems and the Algorithmic Process for Multistep Problem Solving in Python
Solving a multistep word problem in Python and graphing the resulting function
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
Generating and Plotting Partials Sums of a Series in Python
Using Python to calculate the partial sums of a series to estimate the value of the series
Generating and Plotting Sequences in Python
Using Python to numerically, graphically, and analytically find the limit of a sequence