How to Find the Interval of Convergence for this Series?

[alnix]

Homework Statement

serie ((-1)^(n-1)(x-2)^(n-1))/(5^n)

Homework Equations

how to find the interval of convergence for this?

The Attempt at a Solution

 

[tiny-tim]

hi alnix!

(try using the X² button just above the Reply box )

first, simplify it to something you can recognise!

 

[lendav_rott]

Before we can go anywhere, is this what you mean?
$$\frac{(-1)^{n-1}(x-2)^{n-1}}{5^n}$$

 

[HallsofIvy]

The simplest way to determine the radius of convergence is to use the "ratio test".
$$a_n= \frac{(-1)^{n-1}(x- 2)^{n-1}}{5^n}$$
$$a_{n+1}= \frac{(-1)^n(x- 2)^n}{5^{n+1}}$$

What is $\left|\frac{a_{n+1}}{a_n}\right|$? What is the limit of that as n goes to infinity? For what x is that limit less than 1?

 

[Mark44]

To all who have responded in this thread - if someone posts a problem with no effort shown, please use the Report button to let the mentors know.

Alnix, I am closing this thread. Please start a new thread and be sure to show what you have attempted.