Problem 3 (History of infinite series). Peter Gustav Lejeune Dirichlet and Augustin-Louis Cauchy were con- temporaries of Bernhard Riemann (Dirichlet and Riemann from Germany and Cauchy from France). All three were at the center of many debates on the validity and definition of infinite series and limits in general. In 1829, Dirichlet discovered an error in a published paper of Cauchy's, in which Cauchy assumed incorrectly that the Limit Compar- ison Test is valid for all infinite series, not just ones with positive terms. Dirichlet proved this assumption false by considering the following two infinite series:(-1)n OO In and (-1)n 1 + (-1)n n=1 n=1 n n (a) Explain why the first of these series converges, and the second diverges. (b) Calculate lim an , where {an} and {on} are respectively the sequences of terms of the above two series. (c) Summarize (in a verbal sentence or two, or more) Dirichlet's argument against Cauchy. Why do these series form a counter-example to Cauchy's assumption