Finding Convergence Radius & Interval: Solving a Complex Homework Problem

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SUMMARY

The discussion focuses on finding the radius and interval of convergence for the series ∑ ((-1)n (x+3)3n)/(2nlnn) using the Ratio Test. The user initially encountered a negative radius of convergence due to an oversight in applying the limit correctly. By utilizing l'Hôpital's Rule and recognizing the importance of absolute values in the limit, the user corrected their approach, leading to a positive radius of convergence. The key takeaway is the necessity of careful limit evaluation and the correct application of absolute values in convergence tests.

PREREQUISITES
  • Understanding of series convergence tests, specifically the Ratio Test.
  • Familiarity with l'Hôpital's Rule for evaluating limits.
  • Knowledge of absolute convergence and conditional convergence concepts.
  • Basic algebraic manipulation skills, particularly with logarithmic functions.
NEXT STEPS
  • Review the application of the Ratio Test in series convergence.
  • Study l'Hôpital's Rule in depth for limit evaluation.
  • Explore the concepts of absolute and conditional convergence in series.
  • Practice problems involving logarithmic limits and their implications in convergence.
USEFUL FOR

Students studying calculus, particularly those tackling series convergence problems, as well as educators looking for examples of common pitfalls in applying convergence tests.

Grunting7
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Homework Statement


∞
n=3 ∑ ((-1)n (x+3)3n)/(2nlnn)

Find radius of convergence, interval of convergence, values for x which series is: absolutely convergent, conditionally converge or divergence.

Homework Equations

The Attempt at a Solution


I applied the Ratio Test and got

|(x+3)3| lim n--> ∞ (-1(lnn))/(ln(n+1))

Then I used l'Hospitals to get the limit and got -1/2.
So then it's -1/2 * |(x+3)3| = L. Then do the radius and interval stuff.

The problem is that it's -1/2 and the radius can't be negative. I've had one or two similar problems where I keep getting a negative radius. Not sure what I am missing.
 
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The absolute value of -1 is 1. When you're pulling out the (x+3)^3, you have to keep the absolute value on the lnn/ln(n+1) or pull out the -1 with the (x+3)^3. Also I am pretty sure you're missing (1/2) somewhere in your limit.
 
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Wow, did not even know that. Solves the negative radius I was having with the other problems.
Thanks a lot!

Yea, I forgot the 2 in the denominator of the limit.
 

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