Power series and uniform convergence.

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SUMMARY

The discussion centers on the power series (2^n/n)*z^n, which is analyzed for uniform convergence on the interval [-1/3, 1/3]. Participants emphasize the use of the ratio test and root test as standard methods for determining convergence of power series. It is established that if a power series converges on a closed and bounded interval, it converges uniformly within that interval.

PREREQUISITES
  • Understanding of power series and their convergence properties
  • Familiarity with the ratio test and root test for series convergence
  • Knowledge of uniform convergence and its implications
  • Basic calculus concepts related to sequences and series
NEXT STEPS
  • Study the application of the ratio test on power series
  • Explore the root test and its effectiveness in convergence analysis
  • Research uniform convergence and its criteria in detail
  • Examine examples of power series and their convergence on various intervals
USEFUL FOR

Mathematicians, students studying real analysis, and anyone interested in the convergence of power series and uniform convergence principles.

MissC
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Hi.
I have this power serie (2^n/n)*z^n that runs from n=1 to infinity, and I have to show whether it's uniform konvergence on [-1/3, 1/3] or not.

I hope someone can help me with this.
 
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It is fairly standard to use the "ratio test" or "root test" to determine convergence of a power series. And I presume you know that "if a power series converges on a closed and bounded interval, then it converges uniformly there."
 
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