How can I show if the sequence is limited

Question

Answer 1

For conditional convergence of an alternating sign series, you need only show that the limit of a_n --> 0 as n--> infinity.

If you take the limit of your expression for a_n, you see that the numerator tends towards +/- n alternating and that the denominator tends towards (n²)^1/2 or n. Therefore, the limiting behavior is +/- 1 alternating which does not fulfill the convergence criterion. There is a type of summation that recognizes a solution to the series +/- 1 alternating, but it isn't part of intro calculus.

You can formally solve for the limit by dividing top and bottom of a_n by n and taking the limit as n--> infinity. Obviously, this series is also not absolutely convergent (a stricter criterion - it would fail the divergence test if -1 factor removed making it a positive series)

Hope that helps.