How Do You Calculate the Taylor Series for ln(1-x)?

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Discussion Overview

The discussion revolves around calculating the Taylor series for the function f(x) = ln(1-x). Participants explore various aspects including the computation of derivatives, the Maclaurin series, the radius and interval of convergence, and alternative methods for deriving the Taylor series.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Homework-related
  • Debate/contested

Main Points Raised

  • Participants compute the first three derivatives of f(x) and discuss the pattern for the n-th derivative, with some suggesting an expression involving factorials and signs.
  • There is a correction regarding the sign of the second derivative, leading to a discussion about the signs of even and odd derivatives.
  • One participant proposes a formula for the Taylor series based on the derivatives evaluated at zero, while others check for accuracy and correct the coefficients.
  • There is confusion about the general term for the series and the application of the ratio test for determining the radius of convergence.
  • Participants discuss whether the derivatives should be positive or negative and clarify the implications of factorials in the context of derivatives.
  • There is a question about the necessity of expressing the series in a general formula and how to approach the radius of convergence correctly.

Areas of Agreement / Disagreement

Participants generally agree on the method of finding derivatives and constructing the Taylor series, but there are disagreements regarding the signs of the derivatives and the correct application of the ratio test. The discussion remains unresolved on some aspects, particularly the correct general term for the series and the radius of convergence.

Contextual Notes

Limitations include unresolved mathematical steps regarding the general term of the series and the application of the ratio test. There is also ambiguity in the signs of the derivatives and their implications for the Taylor series.

Who May Find This Useful

This discussion may be useful for students and individuals interested in calculus, specifically those learning about Taylor series, derivatives, and convergence tests.

  • #31
d_leet said:
That isn't the correct general term in the first place, and to find the radius of convergence the best way for this problem is either to notice the relation between the series you found and the geometric series or just use the ratio test.

you mean between the series i found in part b and the standard taylor series equation?
 
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  • #32
jkh4 said:
you mean between the series i found in part b and the standard taylor series equation?

What is the standard taylor series equation? The best way to find the radius of convergence is probably to just use the ratio test.
 
  • #33
d_leet said:
What is the standard taylor series equation? The best way to find the radius of convergence is probably to just use the ratio test.

but what i don't understand is, when you do the ratio test

lim |An+1/An|

don't you need an equation involving n? for example like n^x/(n+1)
 
  • #34
jkh4 said:
but what i don't understand is, when you do the ratio test

lim |An+1/An|

don't you need an equation involving n? for example like n^x/(n+1)

Yes you need an equation for the general term of the series an which is one of the things should be found in the first part of this question.
 
  • #35
d_leet said:
Yes you need an equation for the general term of the series an which is one of the things should be found in the first part of this question.

okay, i think i start to get it...what you saying is using the equation derive in part a) right?
 
  • #36
jkh4 said:
okay, i think i start to get it...what you saying is using the equation derive in part a) right?

More or less, yes.
 
  • #37
d_leet said:
More or less, yes.


o...

okay so in general, if we want to know the equation for An to do the ratio test, the An equation we need is derive from the f^(n)(x) pattern right?
 
  • #38
jkh4 said:
o...

okay so in general, if we want to know the equation for An to do the ratio test, the An equation we need is derive from the f^(n)(x) pattern right?

Yes.

/*Extra Chars*/
 
  • #39
d_leet said:
Yes.

/*Extra Chars*/

thank you so much

by the way, one side question, can R be negative?
 
  • #40
jkh4 said:
thank you so much

by the way, one side question, can R be negative?

Your welcome, I'm glad to have been of some help. And no R cannot be negative since it is the limit of an absolute value which is never negative.
 
  • #41
d_leet said:
Your welcome, I'm glad to have been of some help. And no R cannot be negative since it is the limit of an absolute value which is never negative.

okay so in that case -|x| < 1 is wrong right?
 
  • #42
jkh4 said:
okay so in that case -|x| < 1 is wrong right?
Wrong for what? Since |x| is non-negative, -|x| is never positive and so -|x|< 1 is true for all x. If you are talking about using the ratio test to find the radius of convergence, you should not have any negative numbers at all. Go back and check your absolute values again. You want |anxn|, not an|xn|!
|(-1)nx^n| is |x|n, not (-1)n|x|n.
 

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