Interval notation for series converging

[cathy]

Homework Statement

So I solved this problem
∑ (-1)^n (x+5)^n by finding the sum of the series to be 1/(x+6), which was correct. I am stuck on the second part of the problem asking for interval notation.
It says: Determine, in interval notation, the values of x for which the series converges. Use "-I" for negative infinity, "I" for infinity.

2. The attempt at a solution

I know that -x+5 > 1, but how to I turn this into interval notation?
I'm stuck on every one of these problems on my math hw, unfortunately. Please advise if you can. Thanks in advance.

 

[Fredrik]

The standard is to use a "square" bracket (i.e. [ or ]) when the endpoint is included in the interval, and a parenthesis when it's not. For example, [a,b) denotes the set of all extended real numbers* x such that $a\leq x<b$.

*) The set of extended real numbers includes all the real numbers and has exactly two more elements that are usually denoted by $+\infty$ and $-\infty$, but apparently you are supposed to denote them by I and -I.

Another example: Consider the set of all non-negative real numbers, i.e. the set of all real numbers x such that x≥0. Since every real number is less than +∞, we can write 0≤x<+∞ instead of just x>0. So the interval notation for this set would be [0,+∞).

 

[cathy]

I have tried everything, and I cannot seem to get the answer correct.
Can you help me on this one? I really don't know how to do it. I have many similar problems so I will try those once I see how to do it.

 

[Mark44]

Homework Statement

So I solved this problem
∑ (-1)^n (x+5)^n by finding the sum of the series to be 1/(x+6), which was correct. I am stuck on the second part of the problem asking for interval notation.
It says: Determine, in interval notation, the values of x for which the series converges. Use "-I" for negative infinity, "I" for infinity.

2. The attempt at a solution

I know that -x+5 > 1, but how to I turn this into interval notation?

I don't believe that your inequality is correct, so it isn't a matter of turning it into interval notation, but rather, starting with the correct inequality.

Your series is a geometric series, a fact that you used to get the sum of the series. The common ratio, r, is -(x + 5). For such a series to converge, it must be that |r| < 1. In this case, that means that |-(x + 5)| < 1, which is equivalent to |x + 5| < 1. Another way to write this is -1 < x + 5 < 1. Can you solve this inequality?

 

[Fredrik]

I have tried everything, and I cannot seem to get the answer correct.
Can you help me on this one? I really don't know how to do it. I have many similar problems so I will try those once I see how to do it.

You asked about interval notation, and I explained it to you. What kind of answer did you want if not an explanation of interval notation?

I didn't look at the series since you were only asking about the inequality and the notation. I have looked at it now, and I agree with Mark. You need to start with |-(x+5)|<1 and rewrite that as two inequalities a<x<b. The straightforward way to deal with an equality that involves an absolute value is this: |x|<y tells you that if x≥0, then x<y, and that if x≤0, then -x<y.

There's also an easier way based on the notion of distance between two real numbers.

 

[cathy]

Your series is a geometric series, a fact that you used to get the sum of the series. The common ratio, r, is -(x + 5). For such a series to converge, it must be that |r| < 1. In this case, that means that |-(x + 5)| < 1, which is equivalent to |x + 5| < 1. Another way to write this is -1 < x + 5 < 1. Can you solve this inequality?

Would it be (-6, -4)? Ahh. Thank you so much.

Thanks everyone! It helped and I solved the rest of my homework for these! :)

 

[Mark44]

Would it be (-6, -4)? Ahh. Thank you so much.

Thanks everyone! It helped and I solved the rest of my homework for these! :)

Yes, that's it. We're happy to help!