# Power Series and Radius of Convergence

When we deal with power series, the radius of convergence becomes a very important concept. But what exactly *is* the radius of convergence? Let''s find out:

## The radius of convergence defined

We know that a series is "convergent" if it approaches a particular finite value. In other words, a convergent series has a sequence of partial sums that approaches a limit as *n* tends to infinity. We have probably examples of this before -- such as the way certain series approach an asymptote while never actually intersecting with it.

If the series *does* not show this tendency, then we call it "divergent."

## Working with convergent series

How do we know whether this series is convergent:

_{∞}∑(−1)

^{n}

^{n = 0}

s_2 = 1-1+1 =1

s_4 = 1-1+1-1 = 0

Recall that a power series takes the following form:

_{∞}Σc

_{n}(x-a)

^{n}

^{n = 0}

_{n}are numbers. We call the c

_{n}values "coefficients of the series."

The most important thing to remember is that the values of x determine whether a series is divergent or convergent.

We know that whenever x=a, the power series converges.

So what exactly is the radius of convergence? We can envision this as the radius of a "disc" in which the series converges.

If we know that an interval of convergence is the largest in the entire series, we call this (and only this) the interval of convergence. It is centered at the center of the power series. If we need to find the radius of convergence, we simply take half of the interval. Note that this value could be non-negative, real, or even infinite.

If we know that the radius of convergence is centered at a, then it also converges for the real values of x, such that:

r

It also

*diverges*for:

The series converges in the interval

x

a+r

*And:*

*The series diverges for*

a-r

But how do we find the radius of convergence?

One method is the ratio test:

_{∞}∑c

_{n}(x−a)

^{n}

^{n=0}

_{n}+1(x−a)

^{n+1}÷C

_{n}(x−a)

^{n}|

1^{n→∞}

_{∞}∑c

_{n}(x−a)

^{n}

^{n=0}

_{n}(x−a)

^{n})

^{1/n}|

1^{n→∞}

## Practicing our skills

Now let''s use our knowledge to find the radius of convergence for a given power series. Consider the following power series:

_{∞}∑(x−1)

^{n}÷2

^{n}

^{n=0}

^{n+1}/2

^{n+1}÷(x-1)

^{n}/2

^{n})|

This leaves us with:

^{n+1}/2

^{n+1})*(2

^{n}/(x-1)

^{n}|

We can simplify this as:

Now we can take the limits:

lim|(x-1)/2| = |(x-1)/2|

^{n→∞}

The series converges for

*x*such that

1.

In other words:

x-1

2

2

Now we know that the radius of convergence is 2, and the interval of convergence is (-1, 3).

## Topics related to the Power Series and Radius of Convergence

Sum of the First n Terms of an Arithmetic Sequence

## Flashcards covering the Power Series and Radius of Convergence

## Practice tests covering the Power Series and Radius of Convergence

AP Calculus BC Diagnostic Tests

## Pair your student with a tutor who understands the radius of convergence

The radius of convergence is rooted in straightforward mathematical concepts, but putting these concepts together can lead to confusion. Fortunately, a tutor can break down complicated steps into smaller, more manageable chunks. This can make all the difference -- especially if you''re feeling lost during fast-paced classroom lectures. You can also turn to your tutor for help and guidance when you get stuck. Reach out to our Educational Directors today. Varsity Tutors will make sure you get the assistance you need from a qualified math tutor.

- Nuclear Physics Tutors
- Abstract Algebra Tutors
- ERB CTP Courses & Classes
- Louisiana EOC Courses & Classes
- OLSAT Courses & Classes
- CCP-V - Citrix Certified Professional - Virtualization Test Prep
- Geodesy Tutors
- Six Sigma Training
- Florida EOC Assessment Test Prep
- Certified ScrumMaster Courses & Classes
- CLEP Test Prep
- .NET Tutors
- ARM - Associate in Risk Management Test Prep
- Series 53 Courses & Classes
- ARM - Associate in Risk Management Courses & Classes
- Connecticut Bar Exam Courses & Classes
- Actuarial Exam P Courses & Classes
- ASHI - American Society of Home Inspectors Test Prep
- IB Language A: Literature SL Tutors
- California Proficiency Program (CPP) Courses & Classes