Radius of Convergence
When we start to delve into concepts like power series, we encounter many new terms. One such term is the "radius of convergence." How is this concept related to power series? As we will soon find out, the radius of convergence is relatively straightforward to understand, but it''s crucial for the study of power series.
What is the radius of convergence?
Recall that a power series is a type of infinite series. It takes the following form:
${a}_{0}+{a}_{1}x+{a}_{2}{x}^{2}+{a}_{3}{x}^{3}+\dots $We can also write a power series f in the following form (sigma notation):
$f\left(z\right)=\sum _{n=0}^{\infty}{c}_{n}{\left(z-a\right)}^{n}$In this series:
- $a$ is a complex constant and the center of the disk of convergence
- ${c}_{n}$ is the nth complex coefficient
- $z$ is a complex variable
So, what is this "disk of convergence?"
Consider the power series in geometric terms. At the center of the series $\left(a\right)$ , envision a disk. The power series converges for any $x$-value within this disk.
The radius of this disk is the radius of convergence, and it has several interesting properties:
- It is either a non-negative real number or infinite ($\infty$)
- If the radius is positive, the power series converges absolutely and uniformly on any closed subinterval of the interval of convergence
- If $\left|x-a\right|<r$ , the series converges
- If $\left|x-a\right|>r$ , the series diverges
To understand the radius of convergence, we must distinguish between convergent and divergent series. A convergent series has partial sums that approach a specific number, the limit. In contrast, a divergent series does not approach any specific limit; it continues to grow indefinitely.
The interval of convergence is the set of x-values that yield a convergent series when substituted into our power series. The radius of convergence is always half the length of the interval of convergence. This interval essentially represents the diameter of the disk at the center of our power series.
Finding the radius of convergence
But how exactly do we find the radius of convergence?
We can employ something called the Ratio Test. Here''s an example:
- Let ${a}_{n}={c}_{n}{\left(x-a\right)}^{n}$ and ${a}_{n+1}={c}_{n+1}{\left(x-a\right)}^{n+1}$ .
- Simplify the ratio $\frac{{a}_{n+1}}{{a}_{n}}$ to get $\left|\frac{{c}_{n+1}}{{c}_{n}}\left(x-a\right)\right|$ (We take the absolute value of the ratio).
- Calculate the limit of this ratio as $n$ approaches infinity. We denote this limit as $N$ .
The next step is to consider our possibilities based on our calculations:
- If the limit $N$ is zero, the radius of convergence is infinite. This means that the power series converges for all values of $x$ .
- If the limit is $N\left|x-a\right|$ (where $N$ is a constant), then the radius of convergence $R=\frac{1}{N}$ . The interval of convergence includes $\left(a-R,a+R\right)$ and may also include the end-points $x=a-R$ and $x=a+R$ .
- If the limit is infinite, then the radius of convergence is zero, implying the power series converges only at $x=a$ .
Topics related to the Radius of Convergence
Power Series and Radius of Convergence
Flashcards covering the Radius of Convergence
Practice tests covering the Radius of Convergence
AP Calculus BC Diagnostic Tests
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