# 10.3  Convergence of Series
# Art Belmonte, Summer 1996
# These are the shaded gray examples which appear in Section 10.3.
> restart; a:=n->2^(3*n) / (2*n+1)!;

                                      (3 n)
                                     2
                         a := n -> ----------
                                   (2 n + 1)!

> Limit(a(n+1)/a(n), n=infinity); value(");

                                 (3 n + 3)
                                2          (2 n + 1)!
                      lim       ---------------------
                 n -> infinity                (3 n)
                                  (2 n + 3)! 2


                                  0

> Limit(a(n)^(1/n), n=infinity); value(");

                                  /   (3 n)  \(1/n)
                                  |  2       |
                        lim       |----------|
                   n -> infinity  \(2 n + 1)!/


                                  0

> a:=n->ln(n) / n^2; b:=n->1/n^(3/2);
> Limit(a(n)/b(n), n=infinity); value(");

                                     ln(n)
                           a := n -> -----
                                       2
                                      n


                                       1
                            b := n -> ----
                                       3/2
                                      n


                                        ln(n)
                              lim       -----
                         n -> infinity   1/2
                                        n


                                  0

> Int(a(n), n=1..infinity); value(");

                            infinity
                           /
                          |          ln(n)
                          |          ----- dn
                          |            2
                         /            n
                           1


                                  1

>