The center of convergence of a power series is the anchor point around which the series is defined, often denoted as 'a' in the general form of a power series: ∑ cₙ(x - a)ⁿ.
Understanding the Center of Convergence
A power series is an infinite series of the form:
∑ cₙ(x - a)ⁿ = c₀ + c₁(x - a) + c₂(x - a)² + c₃(x - a)³ + ...
where:
- x is a variable.
- cₙ are the coefficients (constants).
- a is the center of the power series.
The center of convergence, 'a', is crucial because it dictates where the power series is "centered". The power series converges for values of x within a certain interval around 'a', known as the interval of convergence.
Relationship to the Interval and Radius of Convergence
- Interval of Convergence: The set of all x values for which the power series converges.
- Radius of Convergence (R): A non-negative real number or ∞ such that the series converges if |x - a| < R and diverges if |x - a| > R.
The interval of convergence is centered at 'a'. If the radius of convergence is R, then the interval of convergence is of the form (a - R, a + R), (a - R, a + R], [a - R, a + R), or [a - R, a + R]. Note that the endpoints (a - R and a + R) require separate analysis to determine if they are included in the interval of convergence.
Example
Consider the power series:
∑ (x - 2)ⁿ / n
Here, the center of convergence, a, is 2. To find the interval of convergence, you'd typically use the ratio test or the root test. The interval will be centered around 2.
Importance
Identifying the center of convergence is the first step in analyzing the convergence behavior of a power series. It is the reference point from which the interval of convergence is determined.
