# Interval Convergence & Function of Alternating Series (-1/3)^n (x-2)^n

• Autunmsky
In summary, the series (to infinity; n=0) (-1/3)^n (x-2)^n has a convergence interval of x∈(1,5) and converges to the function f(x) = 1/(3-x) over this interval. The series can be tested for convergence using the Ratio Test or by checking if a_n+1 ≤ a_n and lim a_n → 0.
Autunmsky
Consider the series (to infinity; n=0) (-1/3)^n (x-2)^n

Find the intercal of convergence for this series.

To what function does this series converage over this interval?

I know this is an alternating series...I just don't know how to go about it. Thanks for you help.

** Should I do this like a partial sum? Do I just keep multiplying them together until they reach Zero *because then it has converaged? **

Last edited:
Autunmsky said:
Consider the series (to infinity; n=0) (-1/3)^n (x-2)^n

Find the intercal of convergence for this series.

To what function does this series converage over this interval?

I know this is an alternating series...I just don't know how to go about it. Thanks for you help.

** Should I do this like a partial sum? Do I just keep multiplying them together until they reach Zero *because then it has converaged? **
It might be helpful to write this series as
$$\sum_{n = 0}^{\infty} (-1)^n \left(\frac{x - 2}{3}\right)^n$$

For some values of x, this is an alternating series, but for others, it's not.
What theorems do you know for determining whether a series converges?

a $$_{n+1}$$$$\leq$$ for all n

lim$$_{n\rightarrow\infty}$$ a$$_{n}$$ = 0

**sorry those are supposed to be lower subscripts**

You're still thinking that this is an alternating series. For some values of x (such as x = 0), it's NOT an alternating series.

Do you know any tests other than the alternating series test?

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Ratio Test so...lim |a $$_{n+1}$$| / |a$$_{n}$$|

OK, so what do you get if you use the Ratio Test?

## 1. What is an alternating series?

An alternating series is a mathematical series where the terms alternate between positive and negative values.

## 2. What is the interval of convergence for (-1/3)^n (x-2)^n?

The interval of convergence for this series is [-3, 1]. This means that the series will converge for all values of x within this interval.

## 3. How do you determine if an alternating series is convergent or divergent?

To determine if an alternating series is convergent or divergent, you can use the Alternating Series Test. This test states that if the absolute value of the terms in the series decreases and approaches 0 as n approaches infinity, then the series is convergent. If the terms do not approach 0, then the series is divergent.

## 4. Can you use the Alternating Series Test to prove that (-1/3)^n (x-2)^n is convergent?

Yes, you can use the Alternating Series Test to prove that this series is convergent. By evaluating the absolute value of the terms, you can see that they decrease and approach 0 as n approaches infinity, making the series convergent.

## 5. Can this series be used to find the value of x?

No, this series cannot be used to find the value of x. It is only used to determine if the series is convergent or divergent within the given interval. To find the value of x, you would need additional information or equations.

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