Evaluating endpoints for conditionally or absolute convergence

**Mondy** · September 18th 2010, 09:01 AM

Q1)
(x^n) / ((2^n)*ln(n)) = x is on the interval of -2 to 2. All of this is correct so far, however, I trouble evaluating the endpoints to see if they conditionally/absolute converges or diverges. According to the book answer, -2 conditionally converges, which I dont quite understand.

My question is, how do I evaluate the endpoints for convergence/divergence? I plugged the numbers in, but dont quite grasp how to proceed from there.

**Plato** · September 18th 2010, 09:33 AM

If $S(x) = \sum\limits_n {\frac{{x^n }}{{2^n \ln (n)}}}$ then $S(2) = \sum\limits_n {\frac{1}{{\ln (n)}}}$ clearly diverges.

Whereas, $S( - 2) = \sum\limits_n {\frac{{\left( { - 1} \right)^n }}{{\ln (n)}}}$ converges by the alternating series test.

Does this help?

**Mondy** · September 19th 2010, 09:13 AM

Thank you! I understand it now.

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