Interval of Convergence for a Series

[calculusisfun]

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?

 

[Dick]

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.

 

[calculusisfun]

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?

 

[Dick]

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?

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.