# Interval of Convergence

1. Aug 8, 2010

### xllGoliathllx

I got a summer homework packet that covers all of last year's curriculum (Pre-Calc) and the first three chapters of next year's curriculum (Calc). I have no idea what I'm really doing with the calc stuff. I'm trying to teach myself sequences, limits, and derivatives. It was going pretty smoothly until this problem:

Find the interval of convergence for: 1 + 4x + 16x^2 + ...

A) x < 1/4
B) -1/4 < x < 1/4
C) 0< x < 1
D) x > -1/4

I'm sure this problem is ridiculously easy, but I can't seem to find anything to teach myself how to do it.

Sorry if I posted this in the wrong section.

2. Aug 8, 2010

### vela

Staff Emeritus
First, identify what type of series that is. You should know the condition under which that type of series converges. Apply that requirement to this problem and you'll have your answer.

3. Aug 8, 2010

### xllGoliathllx

One problem... I've never learned about series. From what I've read, I would say that is a power series. Is that right?

4. Aug 8, 2010

### vela

Staff Emeritus
5. Aug 8, 2010

### xllGoliathllx

I might have seen it before and just forgot about it. Actually come to think of it, I learned about sequences, summation notation, etc last year. I've never seen some of these terms though. I'll take a look at that page and see if I can figure it out.

Thanks for the help,

Matt

6. Aug 8, 2010

### xllGoliathllx

I think 0 < x < 1 makes the most sense. ???

7. Aug 8, 2010

### vela

Staff Emeritus
Nope. If you have a geometric series 1+r+r2+..., it converges if |r|<1. So what is r, the ratio between successive terms, for the series you've been given? Plug that into the inequality and solve for x.

8. Aug 8, 2010

### xllGoliathllx

Common ratio is 4.

Since the |4| is not less than 1, the series doesn't converge??? I'm lost...

Sorry.... I'm actually a pretty smart person when high school teachers actually teach before they give assignments.

It's just 1 question out of 140, so I don't care any more. You don't have to help me... I know it must feel like trying to teach a rock. lol

9. Aug 8, 2010

### vela

Staff Emeritus
Don't give up yet. It's pretty straightforward.

The series is 1+(4x)+(4x)2+.... Compare that to the form of the generic geometric series 1+r+r2+... You can see that r=4x. So...

10. Aug 8, 2010

### xllGoliathllx

lol.... I'm an idiot.

4x<1
x<1/4

Thanks again for the help :),

Matt

11. Aug 8, 2010

### vela

Staff Emeritus
Don't forget the absolute value! It's |r|<1, not r<1.

12. Aug 9, 2010

### HallsofIvy

Staff Emeritus
And so the interval of convergence is
$$-\frac{1}{4}\le x< \frac{1}{4}$$