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

[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? **

 

[Mark44]

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?

 

[Autunmsky]

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

lim_{n\rightarrow\infty} a_{n} = 0

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

 

[Mark44]

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?

 

[Autunmsky]

Ratio Test so...lim |a _{<span style="font-size: 9px">n+1}</span>| / |a_{<span style="font-size: 9px">n}</span>|

 

[Mark44]

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