Power series (expansion series)

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Homework Help Overview

The discussion revolves around the power series expansion of the logarithmic function, specifically ln(1+x). Participants are exploring the conditions under which the series converges, particularly focusing on the values of x that are permissible.

Discussion Character

  • Exploratory, Assumption checking, Conceptual clarification

Approaches and Questions Raised

  • Participants are questioning why the condition for x is restricted to the interval (-1, 1], with specific inquiries about the behavior of the series at x = -1 and |x| > 1. They are comparing the series to its intended representation and discussing the implications of undefined values.

Discussion Status

The discussion is active, with participants raising questions and exploring the implications of the series' convergence. Some guidance has been offered regarding the relationship between the series and the function it represents, but there is no explicit consensus on the conclusions drawn.

Contextual Notes

Participants have noted that the series is only valid for a limited range of x values due to the nature of the logarithmic function, and there are references to external resources for further exploration.

delsoo
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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

 
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Simon Bridge said:
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?
 
Last edited:
...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).
 
referring to my textbook, if x=1 , ln(1+X) converges, can you show me how it can converges please?
 
...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.
 
Last edited:

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