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.
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:
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:
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.
But how exactly do we find the radius of convergence?
We can employ something called the Ratio Test. Here''s an example:
The next step is to consider our possibilities based on our calculations:
Power Series and Radius of Convergence
AP Calculus BC Diagnostic Tests
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