What is the difference between `convergence radius` and `convergencee interval`?
2,978
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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Author by
Billie
Updated on August 01, 2022Comments
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Billie over 1 year
I have a power series $ \sum^\infty_0 = a_nx^n $ , and I have to find the
convergence interval
andconvergence radius
.The convergence radius is $\lim_{n \to \infty} \frac{1}{\sqrt[n]a_n} $, but what is the
convergence interval
? -
Billie almost 10 yearsSo basicly is it the same?
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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.
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Billie almost 10 years@CameronBuie In case the $R = 0$ , then what is the convergence interval? (c, c) = $\phi$
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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.