Interval of Convergence of Power Series with Square Root

In summary, the conversation revolved around finding the interval of convergence for the series $$\sum_{{\rm n}=0}^\infty \left (-\sqrt x \right )^n$$ and using the root test to determine the interval. The questioner initially thought the interval was from 0 to 1, but Wolfram gave -1 to 1, citing the absolute value going inside the square root. The expert clarified that for real values of x, x<0 is not in the domain.
  • #1
I'm trying to find the answer to a question similar to this posted it earlier but in the wrong section I think and not explained well.
$$
\sum_{{\rm n}=0}^\infty \left (-\sqrt x \right )^n \ \ \rm ?$$
Find the interval of convergence?
I tried using the root test and got from 0 to 1 but when I checked with Wolfram and it gave from -1 to 1 saying the absolute value would go inside the square root. Am I doing something wrong or is Wolfram making a mistake. Thanks in advance for any help.
 
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  • #2
Satirical T-rex said:
I'm trying to find the answer to a question similar to this posted it earlier but in the wrong section I think and not explained well.
$$
\sum_{{\rm n}=0}^\infty \left (-\sqrt x \right )^n \ \ \rm ?$$
Find the interval of convergence?
I tried using the root test and got from 0 to 1 but when I checked with Wolfram and it gave from -1 to 1 saying the absolute value would go inside the square root. Am I doing something wrong or is Wolfram making a mistake. Thanks in advance for any help.
Assuming ##x## is real, ##x<0## is not in the domain.
 
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Likes Satirical T-rex
  • #3
Thank you for the help in clearing up my confusion.
 

1. What is the interval of convergence for a power series with a square root?

The interval of convergence for a power series with a square root is the range of values for which the series will converge and give a valid result. It is usually expressed in terms of x-values, and can be found by using the ratio test or the root test.

2. How do you determine the interval of convergence for a power series with a square root?

The interval of convergence can be determined by using the ratio test or the root test. These tests involve taking the limit of the absolute value of the ratio or root of successive terms in the series. If the limit is less than 1, the series will converge, and the x-values within that range will be part of the interval of convergence.

3. What happens if the limit in the ratio or root test for a power series with a square root is equal to 1?

If the limit is equal to 1, the test is inconclusive and further methods may need to be used to determine the interval of convergence. This could include using other convergence tests, manipulating the series, or considering the behavior of the function at the endpoints of the interval.

4. Can a power series with a square root have an infinite interval of convergence?

Yes, a power series with a square root can have an infinite interval of convergence. This means that the series will converge for all real values of x. This can occur if the ratio or root test results in a limit of 0, indicating that the series will converge for all values of x.

5. How does the presence of a square root affect the convergence of a power series?

The presence of a square root can affect the convergence of a power series in a few ways. If the series has a square root term or is raised to a fractional power, it may have a smaller interval of convergence compared to a series without these terms. Additionally, the behavior of the function at the endpoints of the interval of convergence may be different due to the square root, and this should be considered when determining the convergence of the series.

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