The radius of convergence is one of the continuing topics of the Power series. It explains the behavior of certain sequences and whether they will converge or diverge. Before going deep into this topic, it is essential to have a good knowledge of the power series, which mainly comprises the Taylor and McLaurin series. You can check any of the links to understand these topics.
However, if you already have a good understanding of power series, then the radius of convergence will be relatively simple to understand.
In the Power series, the radius of convergence is the biggest disk/point allocated at the center series & is where the entire series converges. The radius can only be a non-negative value, 0 or ∞.
Understanding the radius of convergence is essential for penetrating advanced aspects of the power series, so students must fully follow the instructions outlined in this piece.
What is The Radius of Convergence?
The radius of convergence of a power series is the center value or point to which the series converges. The power radius which is represented by r is either a non-negative number or ∞
Mathematically, the radius of convergence is defined as follows.
Where
a is a real number constant, and the center of the disk of the series convergence
cn is the n-th real coefficient of the and
x is a real variable
then the radius o convergence (represented as r) of the above power series is a non-negative value or ∞ that the series converges to whenever
and diverges whenever
The series may converge or diverge whenever
Explaining The Radius of Convergence
Now, based on the definition of power series, students know that the series below.
cn is the coefficient of the increasing sequence and changes according to the nth value being considered. a, on the other hand, is a constant when equal to zero will generate the McLaurin series.
Now a series is generally said to converge if the additional nth term seems to be approaching a particular value. For the radius of convergence, that value is the r, which is what students need to determine when trying to solve their respective calculations.
How To Calculate The Radius of Convergence?
It is quite easy to calculate the radius of convergence. Students will need to use a Ratio test table to know the value of r. The relevant values of the test will be made available as we consider the different steps of finding the ratio. The table only consists of three values for r, so it is easy to learn. With that out of the way, below are the steps to compute the radius of convergence once given the power series, which will be in the form
Step 5: Simplify the ratio and determine r based on the three values of r in the Ratio test table listed below
| Lim value of the absolute ratio as n → ∞ | r value |
| If the limit of the series equals 0 | r will be Infinite because it means the power series converges for all x values. |
| If the limit of the series equals ∞ | r will equal 0 because the power series converges only at x = a. |
| If the limit of the series ends up in a form such that it is equal to N.|x – a|; with N being a natural number | r = 1/N |
Examples On Radius of Convergence
Consider the Power series below
Now determine the radius of convergence
Solution
Now we only need to implement the steps in the order arranged to determine the radius of convergence which is represented as r
Since the terms are to be expressed in absolute based on the absolute value sign, then the minus sign (-) cannot exist as such; the real term would be
Now, we seem to be at a crossroads at eqn (1). So the best step is to take a look at our r table and see if it can give us an idea of what to do
| Lim value of the absolute ratio as n → ∞ | r value |
| If the limit of the series equals 0 | r will be Infinite because it means the power series converges for all x values. |
| If the limit of the series equals |
r will equal 0 because the power series converges only at x = a. |
| If the limit of the series ends up in a form such that it is equal to N.|x – a|; with N being a natural number | r = 1/N |
- Now looking at eqn (1) it is almost impossible for us to simplify it such that it will become 0. So our r cannot be infinite.
- Also, it is very unlikely that we can make it equal ∞. So our r cannot be equal to 0 either.
- Now for the third row of the table, it is stated whenever we can express the limit in a form N.|x – a| where N is a natural number, then our radius of convergence will be r = 1/N. Now it is more likely for us to achieve this with eqn (1) than with the other two options. Actually, as a rule of thumb, it is imperative to determine whether r can be a positive real value before trying to decide whether or not it is 0 or ∞.
Now back to our Eqn (1)
Now a careful look will show that the eqn (2) is very similar to N.|x – a| where
N = ¼
So this means that our r = 1/N = 1 / ¼ =1 x 4/1 = 4
Therefore, our radius of convergence = 4, and it is the point at which the power series converges
There are two points to note in our calculation above:
- It is essential to find real number solutions to the radius of convergence. It is until none has been confirmed to exist before students can settle for 0 or ∞.
- The rules of absolute value that changes the minus sign (-) to + must be respected
Conclusion
Understanding the ideology of the radius of convergence is more important than the mathematical computation itself in the long term. Infinite series will either converge or diverge. Through the radius of convergence, students can determine whether their power series follows a pattern and will either tend to zero, a number or infinity.
Students can continue to practice different examples on the topic. Generally, math assignments and tests will not churn out heavy mathematical computations for questions regarding the radius of convergence. The example solved in this piece is carefully selected to drive intense understanding so students can easily solve other problems on the radius of convergence.
