Interval of Convergence for a Series

In summary, the person is stuck at finding the interval of convergence for an inequality. They have reached the point of -4 < x^2 < 4 and are unsure how to proceed. Another person explains that the inequality can be solved by replacing the imaginary number with 0, giving the solution of -2<x<2. The person is grateful for the prompt response and understands the solution partially.
  • #1
calculusisfun
31
0

Homework Statement


Ok, so I don't need help with this part, I just got stuck at the following step when attempting to find the interval of convergence:

The Attempt at a Solution



I got here:

-4 < x^2 < 4

So, I need to solve this inequality. But can I? How can I take the square root of negative 4? And if this isn't possible to solve, what is the interval of convergence?
 
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  • #2
The inequality is perfectly easy to solve. It's -2<x<2. For every real number x^2 is greater than -4, so you don't even care about that limit.
 
Last edited:
  • #3
Thanks for the prompt response.

Okay, I understand what you're saying partially.

But to solve the inequality -4 < x^2 < 4, wouldn't you take the square root of both sides of the inequality to get the following:

root(-4) < x < root(4)

And root(-4) is an imaginary number is it not? Which would mean what for the interval of convergence? o_O
 
  • #4
calculusisfun said:
Thanks for the prompt response.

Okay, I understand what you're saying partially.

But to solve the inequality -4 < x^2 < 4, wouldn't you take the square root of both sides of the inequality to get the following:

root(-4) < x < root(4)

And root(-4) is an imaginary number is it not? Which would mean what for the interval of convergence? o_O

Yes, sqrt(-4) is imaginary, but who cares? You aren't looking for imaginary solutions. You want real solutions. Just replace it with 0<=x^2<4.
 

1. What is an interval of convergence for a series?

The interval of convergence for a series is the range of values for which the series will converge, or have a finite sum. Outside of this interval, the series will either diverge or have an infinite sum.

2. How is the interval of convergence determined?

The interval of convergence is determined by the ratio or root test, which involves taking the limit as n approaches infinity of the absolute value of the ratio of successive terms in the series. If the limit is less than 1, the series will converge within a certain interval. If the limit is greater than 1, the series will diverge. If the limit is exactly 1, further tests may be needed to determine the convergence of the series.

3. Can the interval of convergence be infinite?

Yes, the interval of convergence can be infinite, meaning the series will converge for all real values of x. This is often the case for geometric series or power series centered at a particular point.

4. What happens if the value of x is outside the interval of convergence?

If x is outside the interval of convergence, the series will either diverge or have an infinite sum. This means that the series cannot be evaluated at that particular value of x.

5. Can the interval of convergence change for different series with the same general form?

Yes, the interval of convergence can vary for different series with the same general form. This is because the coefficients or powers of the terms in the series may affect the convergence or divergence of the series. It is important to check the interval of convergence for each individual series rather than assuming it will be the same for all series with the same general form.

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