Is it possible for radius of convergence to be negative?

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SUMMARY

The radius of convergence (R) cannot be negative. In the context of power series, R is defined as the distance from the center of convergence to the nearest singularity in the complex plane. The discussion highlights that if the inequality -|x| < 1 holds, it implies that x can take any value, which does not support the notion of a negative radius of convergence. Reference to standard calculus texts confirms this definition and clarifies the misunderstanding surrounding the concept.

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jkh4
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is it possible for "R" (radius of convergence) to be negative?

is it possible for "R" (radius of convergence) to be negative?

for example: -|x|<1 and R=-1?
 
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What does the definition say?
 
jkh4 said:
is it possible for "R" (radius of convergence) to be negative?

for example: -|x|<1 and R=-1?

If -|x|< 1,then x can be any number. So it's not a very useful thing to say
 
As Hurkyl said, take any Calculus book and look up the definition of "radius of convergence"!
 

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