Interval of Convergence
Definition
The interval of convergence is the range of x-values for which a power series converges, meaning it approaches a finite value.
Analogy
Imagine you have a bowl of ice cream. The interval of convergence is like the temperature range in which the ice cream stays solid and doesn't melt. If it's too hot or too cold, the ice cream will either melt or freeze completely.
Related terms
Radius of Convergence: The radius of convergence is the distance from the center point to the edge of the interval of convergence.
Power Series: A power series is an infinite sum of terms that involve powers (exponents) of a variable.
Convergent Series: A convergent series is a sequence whose partial sums approach a finite limit as more terms are added.
"Interval of Convergence" appears in:
Practice Questions (9)
Which test can be used to determine the interval of convergence of a power series?
Which of the following statements is true about the interval of convergence of a power series?
If the radius of convergence of a power series centered at 0 is 5, what can be said about the interval of convergence?
What is the interval of convergence of the power series represented by the function f(x) = ∑(n=0 to ∞) (2x)^n/n^2?
If the interval of convergence for a power series is (-4, 6), what can be said about the radius of convergence?
If a power series has a radius of convergence of 0, what can be said about the interval of convergence?
What is the interval of convergence of the power series represented by the function f(x) = ∑(n=1 to ∞) (x-2)^n/n^3?
If the interval of convergence for a power series is (-1, 5), what can be said about the radius of convergence?
Which of the following statements is true regarding the radius and interval of convergence?
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