Alternating series test resulting in conditional convergence: sum (-1)^n*n/sqrt(n^3+1).

Описание: We test the convergence of an alternating series (-1)^n*n/sqrt(n^3+1), starting with the test for absolute convergence (because if we can show absolute convergence, it automatically implies ordinary convergence of the series).

The series of absolute values approaches the terms of the divergent p-series 1/sqrt(n) in the large n limit, so we perform a limit comparison test and find that our series of absolute values diverges.

So we've failed the test for absolute convergence, which means we should look at the alternating series test in order to decide whether or not our series is conditionally convergent.

For the first part of the alternating series test, we need to show that the terms of the series approach zero in the large n limit (where we are just looking at the magnitude of the terms, not including the alternating sign). This is a quick limit, and we show that it approaches zero.

The second part of the alternating series test is usually the difficult part: we need to show the a_n's are getting smaller, more precisely a_n+1 less than or equal to a_n for all n. Given our expression for a_n, this isn't obvious, so we need to use the derivative of a_n to show that the terms are decreasing. We take the derivative using the quotient and chain rules, and we show that the derivative is negative when n is larger than 1, so we satisfy the second part of the alternating series test using the derivative.

Since the alternating series converges, but the series of absolute values diverges, we conclude that our original alternating series converges conditionally.