Radius of Convergence for Series with (-1)^(n-1) and x^n Terms

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SUMMARY

The radius of convergence for the series \(\sum_{n=1}^{\infty} \frac{(-1)^{n-1} x^n}{n}\) is determined using the ratio test. The series converges absolutely within its radius of convergence, which is found by applying the ratio test to the absolute values of the terms. The series can be recognized as related to the function \(f(x) = \ln(x+1)\), with the derivative \(f'(x) = \frac{1}{1+x}\) indicating that the series converges to this logarithmic function at the right endpoint.

PREREQUISITES
  • Understanding of power series and their convergence
  • Familiarity with the ratio test for series convergence
  • Knowledge of derivatives and their relationship to series
  • Basic concepts of logarithmic functions
NEXT STEPS
  • Learn about the ratio test for series convergence in detail
  • Study the properties of logarithmic functions and their derivatives
  • Explore Abel's Theorem and its applications in series
  • Investigate geometric series and their convergence criteria
USEFUL FOR

Students studying calculus, particularly those focusing on series convergence, as well as educators teaching series and their properties.

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Homework Statement



Find the radius of convergence of \sum n=1 to ∞ of [(-1)^(n-1) x^(n)/(n)] and give a formula for the value of the series at the right hand endpoint.

Homework Equations




The Attempt at a Solution



Not really sure how to start this. I know I'm supposed to use the root test or the ratio test and then probably Abel's Thm for series. but the n-1 in the series is throwing me off. I've never seen that before. Any help is appreciated.
 
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Your series is just f(x)=x-x^2/2+x^3/3+... Just use the ratio test to find the radius of convergence. To find the actual value at the endpoint you could just ask yourself is you have seen that kind of series before. If it doesn't look familiar, try finding f'(x) and see if that series looks familiar.
 
Dick said:
Your series is just f(x)=x-x^2/2+x^3/3+... Just use the ratio test to find the radius of convergence. To find the actual value at the endpoint you could just ask yourself is you have seen that kind of series before. If it doesn't look familiar, try finding f'(x) and see if that series looks familiar.

So I took f'(x) = 1 - x + x^2 - x^3 + ... so, the endpoint is going to look like x^(n+1)? I need a formula for the value at the right hand endpoint..would that work?
 
No, you have to recognize the series, my point to looking at f'(x) was that it's geometric, it's 1/(1+x). if that's f'(x), what's f(x)?
 
If f'(x) = 1/(1+x) then f(x) = ln(x+1)?
 
ln(x+1)+C, you mean. You have to fix the constant C by finding f(0). But yes, that's the idea.
 
Alright, thank you.
 
tracedinair said:

Homework Statement



Find the radius of convergence of \sum n=1 to ∞ of [(-1)^(n-1) x^(n)/(n)] and give a formula for the value of the series at the right hand endpoint.

Homework Equations




The Attempt at a Solution



Not really sure how to start this. I know I'm supposed to use the root test or the ratio test and then probably Abel's Thm for series. but the n-1 in the series is throwing me off. I've never seen that before. Any help is appreciated.
The ratio test applies to a positive series and a power series converges absolutely inside its "radius of convergence".

Take the absolute value (drop the "(-1)^{n-1}") and then use the ratio test.
 

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