Radius of convergence
A power series (a sum of terms a_n x^n) converges for x with |x| < R, where R is the radius of convergence. Test the ends separately.
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Analogy
A radio signal: it is clear within a certain distance of the tower and lost beyond it.
Example
Ratio of terms: |x| / 2.Converges when |x|/2 < 1.So |x| < 2.Radius R = 2.
In practice
Says for which x a power series gives a valid formula.
Practice
Flashcards
- What is the radius of convergence of the series of x^n/2^n?2.
Questions people ask
How do I find the radius of convergence?
Apply the ratio test to the terms, including the powers of x. Require the limit to be less than 1, and solve for x. For x to the n over 2 to the n, the condition is absolute x over 2 below 1, giving R = 2.
What happens at the endpoints of the interval?
The ratio test says nothing at the ends, so test them separately by substituting the endpoint values. For x equal to 2 in x to the n over 2 to the n, the series is 1 + 1 + 1 and so on, which diverges.