44. Assumptions Matter In 1829, Lejeune Dirichlet pointed out that the great French mathematician Augustin Louis Cauchy made a mistake in a published paper by improperly assuming the Limit Comparison Test to be valid for nonpositive series. Here are Dirichlet's two series: n=1 (-1)" √n n=1 (-1)" √n 1+ (-1)") √√n Explain how they provide a counterexample to the Limit Comparison Test when the series are not assumed to be positive.

For problem 2 part 2: To get your comparison series for LCT, choose the series with leading term over leading term. So you want the series of sqrt{n} over n  "leading term over leading term" then go from there

15. DETAILS Determine whether the Ratio Test is inconclusive (that is, it fails to give a definite answer), conclusive (convergent), or conclusive (divergent) for each series. Work must include the set up of the ratio test and the answer to the limit for each. (a) inconclusive O conclusive (convergent) O conclusive (divergent) (b) O inconclusive O conclusive (convergent) O conclusive (divergent) (-6)" - 1 (c) O inconclusive O conclusive (convergent) O conclusive (divergent)