What is the difference between `convergence radius` and `convergencee interval`?

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The convergence interval is the interval upon which the power series converges.

The radius of convergence (convergence radius) is the radius of this interval.

So for example, the series

$$\sum_{n = 0}^{\infty} x^n$$

converges iff $-1 < x < 1$, so the interval is $(-1, 1)$ and the radius is $1$.

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Billie
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Billie

Updated on August 01, 2022

Comments

  • Billie
    Billie over 1 year

    I have a power series $ \sum^\infty_0 = a_nx^n $ , and I have to find the convergence interval and convergence radius.

    The convergence radius is $\lim_{n \to \infty} \frac{1}{\sqrt[n]a_n} $, but what is the convergence interval?

  • Billie
    Billie almost 10 years
    So basicly is it the same?
  • Cameron Buie
    Cameron Buie almost 10 years
    @user1798362: No, but they are related. The convergence radius is half the length of the convergence interval. More generally, if we have a power series $$\sum_{n=0}^\infty a_n(x-c)^n$$ and its convergence radius is $R=\lim\limits_{n\to\infty}\frac{1}{\sqrt[n]{a_n}},$ then its interval of convergence is $(c-R,c+R),$ possibly including one or both of the endpoints.
  • Billie
    Billie almost 10 years
    @CameronBuie In case the $R = 0$ , then what is the convergence interval? (c, c) = $\phi$
  • Cameron Buie
    Cameron Buie almost 10 years
    @user1798362: In that case, the convergence interval will be $\{c\},$ since any power series trivially converges at its center point.