Radius of Convergence

EXPLORE THIS TOPIC IN the MathWorld Classroom

power series sum^(infty)c_kx^kwill converge only for certain values ofx例如,sum_(k=0)^(infty)x^kconverges for-1<x<1In general, there is always an interval(-R,R)in which apower seriesconverges, and the numberRis called the radius of convergence (while the interval itself is called the interval of convergence). The quantityRis called the radius of convergence because, in the case of a power series with complex coefficients, the values ofx|x|<Rform anopen diskwith radiusR

power series总是converges absolutelywithin its radius of convergence. This can be seen by fixingr=|x|and supposing that there exists asubsequence c_(n_i)such that|c_(n_i)|r^(n_i)unbounded然后power series sumc_nx^ndoes notconverge(in fact, the terms are unbounded) because it fails thelimit testTherefore, forxr=|x|>R, the power series does not converge, where

 c=limsup|c_n^(1/n)|
(1)
 R=1/c,
(2)

limsupdenotes thesupremum limit

Conversely, suppose thatr<RThen for any radiussr<s<R, the termsc_nx^n满足

 |c_nx^n|<(s/R)^n
(3)

Nlarge enough (depending ons). It is sufficient to fix a value forsin betweenrRBecauses/R<1, the power series is dominated by a convergentgeometric series. Hence, thepower seriesconverges absolutely by thelimit comparison test

Wolfram Web Resources

Mathematica »

The #1 tool for creating Demonstrations and anything technical.

Wolfram|Alpha »

Explore anything with the first computational knowledge engine.

Wolfram Demonstrations Project »

Explore thousands of free applications across science, mathematics, engineering, technology, business, art, finance, social sciences, and more.

Computerbasedmath.org »

Join the initiative for modernizing math education.

Online Integral Calculator »

Solve integrals with Wolfram|Alpha.

Step-by-step Solutions »

Walk through homework problems step-by-step from beginning to end. Hints help you try the next step on your own.

Wolfram Problem Generator »

Unlimited random practice problems and answers with built-in Step-by-step solutions. Practice online or make a printable study sheet.

Wolfram Education Portal »

Collection of teaching and learning tools built by Wolfram education experts: dynamic textbook, lesson plans, widgets, interactive Demonstrations, and more.

Wolfram Language »

Knowledge-based programming for everyone.