Is it possible for radius of convergence to be negative?

In summary, the radius of convergence of a power series can be negative, indicating that the series will diverge for all values of x and does not represent a valid function. The radius of convergence is determined by using the ratio test on the coefficients of the series, and a negative radius of convergence means that the series does not have a defined limit as x approaches a certain value. A power series cannot have a negative radius of convergence and still converge, and a negative radius of convergence also affects the interval of convergence, making it empty and indicating that the series does not converge for any value of x.
  • #1
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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  • #2
What does the definition say?
 
  • #3
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
 
  • #4
As Hurkyl said, take any Calculus book and look up the definition of "radius of convergence"!
 

Related to Is it possible for radius of convergence to be negative?

1. Can the radius of convergence be negative for a power series?

Yes, it is possible for the radius of convergence to be negative for a power series. This means that the series will not converge for any value of x and therefore does not represent a valid function.

2. What does a negative radius of convergence indicate?

A negative radius of convergence indicates that the power series diverges for all values of x. This means that the series does not have a defined limit as x approaches a certain value.

3. How is the radius of convergence determined for a power series?

The radius of convergence is determined by using the ratio test on the coefficients of the power series. If the limit of the ratio is a finite number, then the radius of convergence is the reciprocal of that number. If the limit is infinity, then the radius of convergence is zero. If the limit is zero, then the radius of convergence is infinity.

4. Can a power series have a negative radius of convergence and still converge?

No, a power series cannot have a negative radius of convergence and still converge. A negative radius of convergence means that the series diverges for all values of x and therefore does not have a valid limit.

5. How does a negative radius of convergence affect the interval of convergence?

A negative radius of convergence means that the series diverges for all values of x, so the interval of convergence is empty. This means that the power series does not converge for any value of x and does not represent a valid function.

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