What Is the Interval of Convergence for the Series Summation?

[uber_kim]

Homework Statement

Find the radius and the interval of convergence for the series:

Ʃ((-1)ⁿxⁿ)/(4ⁿlnn) with sum n=2 to ∞.

Homework Equations

The Attempt at a Solution

I'm testing for the left and right side of the interval. I've found that c_n=1/(lnn), a_n=1/4, and R=1. I used the alternating series test to check the left side, ((-1)ⁿ)/(ln(n)). It passed the decreasing in magnitude and the terms to 0 test, so converges absolutely. The right side, 1/(lnn) > 1/n, so diverges to infinity by squeeze divergence. So the interval would be [-1,1). I'm pretty confident on everything except for the left and right side convergence. Is it correct?

Thanks!

 

[mighty2000]

Thank you for your post. Your approach to finding the radius and interval of convergence for the given series is correct. However, there are a few minor errors in your calculations. Firstly, the alternating series test is only applicable for series with positive terms, so it should not be used for the left side of the interval. Instead, you can use the ratio test to show that the series converges absolutely for x=-1. For the right side, you are correct in using the squeeze divergence test, but the inequality should be reversed, i.e. 1/(lnn) < 1/n. This is because as n approaches infinity, the denominator (lnn) increases faster than the numerator (1). Therefore, the interval of convergence is [-1,1).

I hope this helps clarify any confusion. Keep up the good work!