Power series (expansion series)

[delsoo]

Homework Statement

hi, for the expansion of power series (logarithmitic series) , ln(1+X) , why the condition for x is between -1 and 1 which x can be 1 but x can't be -1 ?

Homework Equations

The Attempt at a Solution

 

[Simon Bridge]

What happens when x=-1 or |x|>1?
i.e. compare the series with what it is supposed to represent.

note: http://hyperphysics.phy-astr.gsu.edu/hbase/math/lnseries.html

 

[delsoo]

What happens when x=-1 or |x|>1?
i.e. compare the series with what it is supposed to represent.

note: http://hyperphysics.phy-astr.gsu.edu/hbase/math/lnseries.html

IF x--1 , then y=ln(0) , which is undefined , if |x|>1, i would get x less than- 1 and x more than 1 ... sub x value (less than -1 ), i would get ln(-x) which is also undefined, sub x value(more than 1 ), i would get ln (infinity) what does it represent?

 

[Simon Bridge]

...what does it represent?

The power series you refer to is supposed to represent ln(x+1) ... so you have noticed that ln(x+1) is only defined for x>-1. This should tell you part of the answer to your question.

To understand the rest of your question, you have to refer to the series itself - what is the series?
What happens to the series when x=-1? Is this consistent with what happens to ln(x+1)?
What happens to the series when |x|>1? Is that consistent?

What I'm getting at is that the series is only valid for a narrow range of possible x values because those are the only values where the series, summed to infinity, is equal to ln(x+1).

 

[delsoo]

referring to my textbook, if x=1 , ln(1+X) converges, can you show me how it can converges please?

 

[Simon Bridge]

...can you show me how it can converges please?

I can but I won't - that is perilously close to doing work for you, that you are best advised to do yourself.

You should be able to show yourself that it converges: consider what it means for a function to converge.
i.e. where does the power series coincide with the function?

You appear to be ignoring suggestions. If you do not follow suggestions nobody can help you.
You've run into this problem in other threads too.