- #1
alnix
- 3
- 0
Homework Statement
serie ((-1)^(n-1)(x-2)^(n-1))/(5^n)
Homework Equations
how to find the interval of convergence for this?
How to Find the Interval of Convergence for this Series?
- Thread starter alnix
- Start date
-
- Tags
- Convergence Interval
In summary, the conversation is about finding the interval of convergence for the series ((-1)^(n-1)(x-2)^(n-1))/(5^n). The suggested method is to use the ratio test and determine the limit as n goes to infinity for the expression |(a_{n+1})/(a_n)|. There is also a reminder to show effort in future problem posts.
serie ((-1)^(n-1)(x-2)^(n-1))/(5^n)
how to find the interval of convergence for this?
- #2
tiny-tim
Science Advisor
Homework Helper
- 25,839
- 258
hi alnix! 
(try using the X2 button just above the Reply box)
first, simplify it to something you can recognise!
(try using the X2 button just above the Reply box
)first, simplify it to something you can recognise!
- #3
lendav_rott
- 232
- 10
Before we can go anywhere, is this what you mean?
[tex]\frac{(-1)^{n-1}(x-2)^{n-1}}{5^n}[/tex]
[tex]\frac{(-1)^{n-1}(x-2)^{n-1}}{5^n}[/tex]
- #4
HallsofIvy
Science Advisor
Homework Helper
- 42,988
- 981
The simplest way to determine the radius of convergence is to use the "ratio test".
[tex]a_n= \frac{(-1)^{n-1}(x- 2)^{n-1}}{5^n}[/tex]
[tex]a_{n+1}= \frac{(-1)^n(x- 2)^n}{5^{n+1}}[/tex]
What is [itex]\left|\frac{a_{n+1}}{a_n}\right|[/itex]? What is the limit of that as n goes to infinity? For what x is that limit less than 1?
[tex]a_n= \frac{(-1)^{n-1}(x- 2)^{n-1}}{5^n}[/tex]
[tex]a_{n+1}= \frac{(-1)^n(x- 2)^n}{5^{n+1}}[/tex]
What is [itex]\left|\frac{a_{n+1}}{a_n}\right|[/itex]? What is the limit of that as n goes to infinity? For what x is that limit less than 1?
- #5
Mark44
Mentor
- 37,779
- 10,165
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.
Alnix, I am closing this thread. Please start a new thread and be sure to show what you have attempted.
FAQ: How to Find the Interval of Convergence for this Series?
What is an interval of convergence?
An interval of convergence is the range of values for which an infinite series converges. It is typically represented as a range of x-values in which the series will converge to a specific value.
How is the interval of convergence determined?
The interval of convergence is determined by using mathematical tests and techniques such as the ratio test, root test, or comparison test. These tests help determine the range of x-values for which the series converges.
What happens if a value falls outside the interval of convergence?
If a value falls outside the interval of convergence, the series will diverge and will not converge to a specific value. This means that the series does not have a finite sum, and its terms continue to increase or decrease indefinitely.
Does every infinite series have an interval of convergence?
No, not every infinite series has an interval of convergence. Some series may have a finite interval of convergence, while others may have an infinite interval of convergence. There are also some series that do not have an interval of convergence at all and are considered divergent.
Can the interval of convergence change?
Yes, the interval of convergence can change depending on the values of the series or the method used to determine it. For example, if a different test is used to determine the interval of convergence, the resulting range of x-values may be different.