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 infinite series (including all the tests) power series taylor series

### Calculus Topics Listed Alphabetically

Single Variable Calculus

 Absolute Convergence Alternating Series Arc Length Area Under Curves Chain Rule Concavity Conics Conics in Polar Form Conditional Convergence Continuity & Discontinuities Convolution, Laplace Transforms Cosine/Sine Integration Critical Points Cylinder-Shell Method - Volume Integrals Definite Integrals Derivatives Differentials Direct Comparison Test Divergence (nth-Term) Test
 Ellipses (Rectangular Conics) Epsilon-Delta Limit Definition Exponential Derivatives Exponential Growth/Decay Finite Limits First Derivative First Derivative Test Formal Limit Definition Fourier Series Geometric Series Graphing Higher Order Derivatives Hyperbolas (Rectangular Conics) Hyperbolic Derivatives
 Implicit Differentiation Improper Integrals Indeterminate Forms Infinite Limits Infinite Series Infinite Series Table Infinite Series Study Techniques Infinite Series, Choosing a Test Infinite Series Exam Preparation Infinite Series Exam A Inflection Points Initial Value Problems, Laplace Transforms Integral Test Integrals Integration by Partial Fractions Integration By Parts Integration By Substitution Intermediate Value Theorem Interval of Convergence Inverse Function Derivatives Inverse Hyperbolic Derivatives Inverse Trig Derivatives
 Laplace Transforms L'Hôpital's Rule Limit Comparison Test Limits Linear Motion Logarithm Derivatives Logarithmic Differentiation Moments, Center of Mass Mean Value Theorem Normal Lines One-Sided Limits Optimization
 p-Series Parabolas (Rectangular Conics) Parabolas (Polar Conics) Parametric Equations Parametric Curves Parametric Surfaces Pinching Theorem Polar Coordinates Plane Regions, Describing Power Rule Power Series Product Rule
 Quotient Rule Radius of Convergence Ratio Test Related Rates Related Rates Areas Related Rates Distances Related Rates Volumes Remainder & Error Bounds Root Test Secant/Tangent Integration Second Derivative Second Derivative Test Shifting Theorems Sine/Cosine Integration Slope and Tangent Lines Square Wave Surface Area
 Tangent/Secant Integration Taylor/Maclaurin Series Telescoping Series Trig Derivatives Trig Integration Trig Limits Trig Substitution Unit Step Function Unit Impulse Function Volume Integrals Washer-Disc Method - Volume Integrals Work

Multi-Variable Calculus

 Acceleration Vector Arc Length (Vector Functions) Arc Length Function Arc Length Parameter Conservative Vector Fields Cross Product Curl Curvature Cylindrical Coordinates
 Directional Derivatives Divergence (Vector Fields) Divergence Theorem Dot Product Double Integrals - Area & Volume Double Integrals - Polar Coordinates Double Integrals - Rectangular Gradients Green's Theorem
 Lagrange Multipliers Line Integrals Partial Derivatives Partial Integrals Path Integrals Potential Functions Principal Unit Normal Vector
 Spherical Coordinates Stokes' Theorem Surface Integrals Tangent Planes Triple Integrals - Cylindrical Triple Integrals - Rectangular Triple Integrals - Spherical
 Unit Tangent Vector Unit Vectors Vector Fields Vectors Vector Functions Vector Functions Equations

Differential Equations

 Boundary Value Problems Bernoulli Equation Cauchy-Euler Equation Chebyshev's Equation Chemical Concentration Classify Differential Equations Differential Equations Euler's Method Exact Equations Existence and Uniqueness Exponential Growth/Decay
 First Order, Linear Fluids, Mixing Fourier Series Inhomogeneous ODE's Integrating Factors, Exact Integrating Factors, Linear Laplace Transforms, Solve Initial Value Problems Linear, First Order Linear, Second Order Linear Systems
 Partial Differential Equations Polynomial Coefficients Population Dynamics Projectile Motion Reduction of Order Resonance
 Second Order, Linear Separation of Variables Slope Fields Stability Substitution Undetermined Coefficients Variation of Parameters Vibration Wronskian

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17calculus > infinite series > radius/interval of convergence

 Difference Between Radius and Interval of Convergence Using The Ratio Test Radius of Convergence Interval of Convergence Practice

In order to have a complete picture of what is going on with a power series (and Taylor series, since Taylor series is a special case of power series), we need to know the radius and interval of convergence. This is analogous to knowing the domain of a function.

Difference Between Radius and Interval of Convergence

The difference between the radius of convergence and the interval of convergence lies in what information we have about the endpoints.

gives convergence information only between x-values, not AT the x-values

Example: if $$(5,7)$$ is the radius of convergence

the series converges for $$5 < x < 7$$

interval of convergence

gives convergence information inside the interval AND convergence/divergence information at the endpoints

Example: if $$(5,7)$$ is the interval of convergence

the series converges for $$5 < x < 7$$ AND we know that the series diverges at $$x = 5$$ and $$x = 7$$

contains the same information as the radius of convergence as well as what is going on at the endpoints

The interval of convergence is sometimes called the convergence set.

Once the Taylor series or power series is calculated, we use the ratio test to determine the radius convergence and other tests to determine the interval of convergence. Without knowing the radius and interval of convergence, the series is not considered a complete function (This is similar to not knowing the domain of a function. As we say in the page on domain and range, a fully defined function always contains information about the domain, either implicitly or explicitly stated.) In this case, we can't leave off information about where the series converges because the series may not hold for all values of $$x$$.

Using The Ratio Test

The ratio test looks like this. We have a series $$\sum{a_n}$$ with non-zero terms. We calculate the limit $$\displaystyle{\lim_{n \to \infty}{\left| \frac{a_{n+1}}{a_n} \right|} = L}$$.
There are three possible cases for the value $$L$$.
$$L < 1$$: The series converges absolutely.
$$L = 1$$: The ratio test is inconclusive.
$$L > 1$$: The series diverges.

So we use the first case ($$L < 1$$, since we want convergence) and we set up the inequality $$\displaystyle{\lim_{n \to \infty}{\left| \frac{a_{n+1}}{a_n} \right|} < 1} ~~~~~ [ 1 ]$$

Key: Do not drop the absolute values.

To find the radius of convergence, we need to simplify the inequality [1] to the point that we have $$\left| x-a \right| < R$$. This gives the radius of convergence as $$R$$. This seems very simple but you need to be careful of the notation and wording your textbooks. Some textbooks use a small $$r$$. Some textbooks ask for the ratio of convergence in which case you need to give the answer as $$\rho = 1/R$$. This has always seemed kind of strange to me but there must be some reason for it. However, on this site, whenever we talk about the radius of convergence in this context, we will use $$R$$ as defined above.

There are three possible cases for the radius of convergence.

 $$R = 0$$ series converges only at the point $$x = a$$ $$0 < R < \infty$$ series converges within the interval $$R = \infty$$ series converges for all $$x$$

The interesting thing is that we have a strict inequality in $$0 < R < \infty$$ and, because of the definition of the ratio test, we have no idea what happens when $$\left| x-a \right| = R$$. The series could converge or diverge. The ratio test doesn't give us a clue on what happens in that case. That's where we need to find the interval of convergence, which we discuss next.

Interval of Convergence

We use the radius of convergence, $$R$$, to calculate the interval of convergence as follows
$$\begin{array}{rcccl} & & \left| x-a \right| & < & R \\ -R & < & x-a & < & R \\ -R + a & < & x & < & R + a \end{array}$$
So now we have an open interval $$(-R+a, R+a)$$ in which the series converges. Now we need to look at the endpoints to determine what goes on. To do that we substitute each endpoint individually for $$x$$ into the series and then use the other series test to determine convergence or divergence.

Notice when we substitute $$x=-R+a$$ into the $$(x-a)^n$$ term, we end up with $$(-R)^n$$ which can be simplified to $$(-1)^n R^n$$. Now we have an alternating series. So often, the alternating series test can be used to determine convergence or divergence. The point is that, using other tests, we need to definitively determine convergence or divergence at each endpoint. The result will be an open interval, a half-open interval or a closed interval. We call this interval, the interval of convergence.

Notice the difference between the terms radius of convergence and interval of convergence. The radius of convergence gives information about the open interval but says nothing about the endpoints. The interval of convergence includes the radius of convergence AND information about convergence or divergence of the endpoints.

Here is a video clip that explains how to show that a series converges for all x.

### Dr Chris Tisdell - What is a Taylor series? [min-secs]

video by Dr Chris Tisdell

Sometimes you are given the interval of convergence and asked to determine the radius convergence. Here is a short video explaining how to do that.

### PatrickJMT - Radius of Convergence for a Power Series [min-secs]

video by PatrickJMT

### Practice

Conversion Between A-B-C Level (or 1-2-3) and New Numbered Practice Problems

Please note that with this new version of 17calculus, the practice problems have been relabeled but they are MOSTLY in the same order. So, Practice A01 (1) is probably the first basic practice problem, A02 (2) is probably the second basic practice problem, etc. Practice B01 is probably the first intermediate practice problem and so on.

GOT IT. THANKS!

Instructions - - Unless otherwise instructed, determine the radius and interval of convergence for each series.

Basic Problems

$$\displaystyle{ \sum_{n=0}^{\infty}{ \frac{x^n}{n!} } }$$

Problem Statement

$$\displaystyle{ \sum_{n=0}^{\infty}{ \frac{x^n}{n!} } }$$

Solution

In this video, he writes the geometric series in a little different form than we use on this page. He writes it as $$\sum{c_n(x-a)^n}$$.

### 293 video

video by Dr Chris Tisdell

$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{(-1)^n}{n} (x-5)^n } }$$

Problem Statement

$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{(-1)^n}{n} (x-5)^n } }$$

Solution

In this video, he writes the geometric series in a little different form than we use on this page. He writes it as $$\sum{ c_n(x-a)^n }$$.

### 294 video

video by Dr Chris Tisdell

$$\displaystyle{ \sum_{n=1}^{\infty}{ nx^n } }$$

Problem Statement

$$\displaystyle{ \sum_{n=1}^{\infty}{ nx^n } }$$

Solution

In this video, he writes the geometric series in a little different form than we use on this page. He writes it as $$\sum{ c_n (x-a)^n }$$.

### 295 video

video by Dr Chris Tisdell

$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{x^n}{\sqrt{n}} } }$$

Problem Statement

$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{x^n}{\sqrt{n}} } }$$

Solution

### 316 video

video by PatrickJMT

$$\displaystyle{ e^{x^2} = \sum_{n=0}^{\infty}{ \frac{x^{2n}}{n!} } }$$

Problem Statement

$$\displaystyle{ e^{x^2} = \sum_{n=0}^{\infty}{ \frac{x^{2n}}{n!} } }$$

Solution

### 317 video

video by PatrickJMT

$$\displaystyle{ \sum_{n=0}^{\infty}{ n^3 (x-5)^n } }$$

Problem Statement

$$\displaystyle{ \sum_{n=0}^{\infty}{ n^3 (x-5)^n } }$$

Solution

### 344 video

video by PatrickJMT

$$\displaystyle{ \sum{ \left(\frac{x}{2}\right)^n } }$$

Problem Statement

$$\displaystyle{ \sum{ \left(\frac{x}{2}\right)^n } }$$

Solution

### 345 video

video by MIT OCW

$$\displaystyle{ \sum{ \frac{x^n}{n~2^n} } }$$

Problem Statement

$$\displaystyle{ \sum{ \frac{x^n}{n~2^n} } }$$

Solution

### 346 video

video by MIT OCW

$$\displaystyle{ \sum{ \frac{x^{2n}}{(2n)!} } }$$

Problem Statement

$$\displaystyle{ \sum{ \frac{x^{2n}}{(2n)!} } }$$

Solution

### 347 video

video by MIT OCW

$$\displaystyle{ \sum_{n=1}^{\infty}{ (-1)^n n x^n } }$$

Problem Statement

$$\displaystyle{ \sum_{n=1}^{\infty}{ (-1)^n n x^n } }$$

Solution

### 348 video

video by MIP4U

$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{(-1)^n x^n}{n} } }$$

Problem Statement

$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{(-1)^n x^n}{n} } }$$

Solution

### 349 video

video by MIP4U

$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{(x+1)^{2n}}{n!} } }$$

Problem Statement

$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{(x+1)^{2n}}{n!} } }$$

Solution

### 351 video

video by MIP4U

$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{5^n x^n}{n!} } }$$

Problem Statement

$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{5^n x^n}{n!} } }$$

Solution

### 374 video

video by Krista King Math

$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{x^n 2^n}{n!} } }$$

Problem Statement

$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{x^n 2^n}{n!} } }$$

Solution

### 376 video

video by PatrickJMT

Maclaurin series for $$\cos(x^2)$$

Problem Statement

Maclaurin series for $$\cos(x^2)$$

Solution

### 1364 video

video by Krista King Math

$$\displaystyle{ \sum_{k=1}^{\infty}{ \frac{x^k}{3^kk^2} } }$$

Problem Statement

$$\displaystyle{ \sum_{k=1}^{\infty}{ \frac{x^k}{3^kk^2} } }$$

$$[-3,3]$$

Problem Statement

$$\displaystyle{ \sum_{k=1}^{\infty}{ \frac{x^k}{3^kk^2} } }$$

Solution

### 2000 video

video by Dr Chris Tisdell

$$[-3,3]$$

Intermediate Problems

$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{(x-2)^n}{(n+1)3^n} } }$$

Problem Statement

$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{(x-2)^n}{(n+1)3^n} } }$$

Solution

### 350 video

video by MIP4U

$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{(x-4)^n}{\sqrt[3]{n}} } }$$

Problem Statement

$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{(x-4)^n}{\sqrt[3]{n}} } }$$

Solution

### 373 video

video by Krista King Math

$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{n^3 (x+5)^n}{6^n} } }$$

Problem Statement

$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{n^3 (x+5)^n}{6^n} } }$$

Solution

### 375 video

video by PatrickJMT

$$\displaystyle{ \sum_{n=0}^{\infty}{ \frac{(x+2)^n}{n+3} } }$$

Problem Statement

$$\displaystyle{ \sum_{n=0}^{\infty}{ \frac{(x+2)^n}{n+3} } }$$

Solution

### 377 video

video by PatrickJMT

$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{x^n 5^n}{n^n} } }$$

Problem Statement

$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{x^n 5^n}{n^n} } }$$

Solution

### 378 video

video by PatrickJMT

$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{(x-2)^n}{\ln(n+4)} } }$$

Problem Statement

$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{(x-2)^n}{\ln(n+4)} } }$$

Solution

### 379 video

video by PatrickJMT

$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{(x+3)^n}{n^2+2n} } }$$

Problem Statement

$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{(x+3)^n}{n^2+2n} } }$$

Solution

### 380 video

video by PatrickJMT

$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{ (-1)^n (x+1)^n }{ 5^n \sqrt{n} } } }$$

Problem Statement

$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{ (-1)^n (x+1)^n }{ 5^n \sqrt{n} } } }$$

Solution

### 381 video

video by PatrickJMT

$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{ (-1)^n (x+2)^n }{ (n+1)\ln(n+1) } } }$$

Problem Statement

$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{ (-1)^n (x+2)^n }{ (n+1)\ln(n+1) } } }$$

Solution

### 382 video

video by PatrickJMT