# Interval of Convergence

• hpayandah
In summary: It looks more complicated than necessary- it should be obvious that \frac{3^{n+1}- 4}{3^n- 4}is dominated by 3^{n+1}{3^n}= 3 and so its limit is 3. But your result, that the limit is (3/4)|x−1| is correct. Now, what is the answer to your original question?

## Homework Statement

Find the interval of convergence of each of the following
Ʃ$^{∞}_{n=0}$ ($\frac{3^{n}-2^{2}}{2^{2n}}$(x-1)$^{n}$)

## The Attempt at a Solution

Please refer to attachment. All I want to know is that I'm doing this problem right. I have found the interval but haven't plugged the interval back into the equation.

#### Attachments

• 20111125_002.jpg
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You shouldn't break it into two series like that unless you know what you are doing. And combining them into (3/4)(x-1)-(1/4)(x-1) shows you probably don't. Just apply the ratio test to the whole expression.

Dick said:
You shouldn't break it into two series like that unless you know what you are doing. And combining them into (3/4)(x-1)-(1/4)(x-1) shows you probably don't. Just apply the ratio test to the whole expression.

The thing is I don't know what to do with that 2^2 in the equation, it throws me off. I'll re-post a picture this time everything together.

hpayandah said:
The thing is I don't know what to do with that 2^2 in the equation, it throws me off. I'll re-post a picture this time everything together.

One of the factors in your ratio test should be (3^(n+1)-2^2)/(3^n-2^2), right? To find the limit of that as n->infinity, just divide numerator and denominator by 3^n.

Dick said:
One of the factors in your ratio test should be (3^(n+1)-2^2)/(3^n-2^2), right? To find the limit of that as n->infinity, just divide numerator and denominator by 3^n.

I tried your suggestion, however I didn't come to a good end (maybe I did something wrong). Please take a look at the attached file; this is my re-attempt.

#### Attachments

• att.jpg
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It looks more complicated than necessary- it should be obvious that
$$\frac{3^{n+1}- 4}{3^n- 4}$$
is dominated by $3^{n+1}{3^n}= 3$ and so its limit is 3. But your result, that the limit is $(3/4)|x- 1|$ is correct. Now, what is the answer to your original question?

hpayandah said:
I tried your suggestion, however I didn't come to a good end (maybe I did something wrong). Please take a look at the attached file; this is my re-attempt.

Not right. There's a mistake in the long division. 3^(n+1)-3^n isn't 3. And I didn't mean that in my suggestion. Take (3^(n+1)-4)/(3^n-4). Dividing the numerator, 3^(n+1)-4 by 3^n gives you 3^(n+1)/3^n-4/3^n. What's 3^(n+1)/3^n? Now take the limit as n->infinity. Do the same with the denominator.

Dick said:
Not right. There's a mistake in the long division. 3^(n+1)-3^n isn't 3. And I didn't mean that in my suggestion. Take (3^(n+1)-4)/(3^n-4). Dividing the numerator, 3^(n+1)-4 by 3^n gives you 3^(n+1)/3^n-4/3^n. What's 3^(n+1)/3^n? Now take the limit as n->infinity. Do the same with the denominator.
Thank you I got it now.

It looks more complicated than necessary- it should be obvious that
3n+1−43n−4

is dominated by 3n+13n=3 and so its limit is 3. But your result, that the limit is (3/4)|x−1| is correct. Now, what is the answer to your original question?
Thanks

## 1. What is an interval of convergence?

An interval of convergence is a range of values for which a given mathematical series will converge, meaning that the sum of the terms in the series will approach a finite value as the number of terms increases.

## 2. How do you determine the interval of convergence for a series?

To determine the interval of convergence for a series, you can use several methods, such as the Ratio Test, Root Test, or Alternating Series Test. These tests involve evaluating the limit of a sequence or a series and comparing it to known values to determine if the series converges or diverges.

## 3. Can a series have multiple intervals of convergence?

Yes, a series can have multiple intervals of convergence. This occurs when the series has different convergence behaviors at different points in the interval. For example, a power series may have an interval of convergence for which it converges absolutely and another interval for which it converges conditionally.

## 4. What happens if the value falls outside of the interval of convergence?

If a given value falls outside of the interval of convergence, the series will diverge, meaning that the sum of the terms will approach infinity rather than a finite value. This indicates that the series is not a valid representation of the function at that point.

## 5. Can the interval of convergence change for a given series?

Yes, the interval of convergence can change for a given series. This can occur if the series is modified or manipulated in some way, such as by taking a derivative or anti-derivative. These changes can affect the convergence behavior of the series and therefore alter the interval of convergence.