Series Convergence/Divergence Flow Chart
              TEST FOR DIVERGENCE
               Does lim      a =0?                                           NO                                  Pan Diverges
                        n→∞ n
                             YES
              p-SERIES        p                                                               YES               Pan Converges
               Does an = 1=n , n ≥ 1?                     YES            Is p > 1?
                   NO                                                                          NO                Pan Diverges
                                                                                                                 P
              GEOMETRICSERIES                                                                                       ∞ a = a
                                                                                                                    n=1 n     1−r
               Does a =arn−1, n ≥ 1?                      YES           Is |r| < 1?            YES
                      n
                   NO                                                                          NO                Pan Diverges
              ALTERNATING SERIES
                               n
               Does an = (−1) bn or                  YES       Is bn+1 ≤ bn & lim bn = 0?         YES           Pan Converges
                         n−1                                                   n→∞
               an = (−1)     bn, bn ≥ 0?
                   NO
              TELESCOPING SERIES
               Dosubsequenttermscanceloutprevioustermsinthe                       Does                YES           Pan=s
               sum? May have to use partial fractions, properties        YES       lim sn = s
                                                                                  n→∞
               of logarithms, etc. to put into appropriate form.                  s finite?           NO          Pan Diverges
                        NO
                                                                                                                 P
              TAYLORSERIES                                                                                         ∞ a =f(x)
                                                                                                    YES            n=0 n
                           f(n)(a)      n            YES     Is x in interval of convergence?
               Does an =     n!  (x−a) ?                                                             NO          Pa Diverges
                                                                                                                     n
                              NO
            Try one or more of the following tests:
              COMPARISONTEST                            YES           Is 0 ≤ an ≤ bn?          YES              Pan Converges
               Pick {bn}. Does Pbn converge?             NO NO
                                                                      Is 0 ≤ bn ≤ an?           YES              Pan Diverges
              LIMIT COMPARISON TEST
                                     a                                   ∞                                      Pan Converges
                                      n
               Pick {bn}. Does lim b = c > 0                            X                        YES
                                n→∞ n                  YES        Does      bn converge?
               c finite & an;bn > 0?                                     n=1                       NO             Pan Diverges
              INTEGRALTEST
                                                                                                              P
                                                                                                                 ∞ a Converges
               Does an = f(n), f(x) is contin-                       Z ∞                          YES            n=a n
               uous, positive & decreasing on        YES        Does      f(x)dx converge?
               [a;∞)?                                                  a                            NO           Pan Diverges
              RATIO TEST                                                                      YES            Pan Abs. Conv.
                                                                            a   
               Is lim     |a    =a | 6= 1?                          Is lim    n+1 <1?
                     n→∞ n+1 n                         YES                   a  
                                                                       n→∞     n
                                                                                                  NO             Pan Diverges
              ROOTTEST                                                      p                                  Pan Abs. Conv.
                          p                                                 n                   YES
               Is lim      n |a | 6= 1?                YES          Is lim    |an| < 1?
                     n→∞       n                                      n→∞
                                                                                                  NO             Pan Diverges
               Problems 1-38 from Stewart’s Calculus, page 784
                   ∞ 2                                          ∞                                            ∞
               1. X n −1                                  14. Xsin(n)                                   27. X kln(k)
                        2                                                                                               3
                  n=1 n +n                                     n=1                                          k=1 (k +1)
                   ∞                                            ∞
                  Xn−1                                         X              n!                             ∞ 1=n
               2.     n2 +n                               15.      2·5·8·····(3n+2)                     28. Xe 2
                  n=1                                          n=0                                          n=1 n
                   ∞                                            ∞ 2
                  X 1                                          Xn +1                                         ∞      −1
               3. n=1 n2 +n                               16. n=1 n3 +1                                 29. Xtan√(n)
                                                                                                            n=1    n n
                   ∞                                            ∞
                  X n−1n−1                                     X n1=n                                        ∞         √
               4.     (−1)     n2 +n                      17.      (−1) 2                                   X j j
                  n=1                                          n=1                                      30.     (−1) j +5
                                                                                                            j=1
                   ∞        n+1                                 ∞        n−1
                  X(−3)                                        X(−1)
               5.         3n                              18.       √                                        ∞      k
                  n=1    2                                     n=2    n−1                               31. X k5 k
                                                                                                                3 +4
                   ∞                                          ∞
                                  n                                                                         k=1
                  X 3n                                         X nln(n)
               6.       1+8n                              19.      (−1)    √                                 ∞
                                                                             n                                       n
                  n=1                                          n=1                                      32. X(2n)
                                                                                                                  n2n
                   ∞                                            ∞
               7. X p1                                    20. Xk+5                                          n=1
                      n ln(n)                                        5k
                                                                                                             ∞
                  n=2                                          k=1                                          Xsin(1=n)
                   ∞     k                                      ∞                                       33.        √
                  X                                                      2n                                          n
               8.       2 k!                              21. X (−2)                                        n=1
                      (k +2)!                                         nn
                  k=1                                          n=1                                           ∞
                   ∞                                            ∞     √                                 34. X          1 2
               9. Xk2e−k                                  22. X         n2 −1                               n=1 n+ncos (n)
                                                                   n3 +2n2+5
                  k=1                                          n=1                                                          2
                                                                                                             ∞          
                                                                                                                          n
                   ∞                                            ∞                                           X n
                  X 3                                          X                                        35.
             10.      n2e−n                               23.      tan(1=n)                                       n+1
                                                                                                            n=1
                  n=1                                          n=1
                                                                                                             ∞
                   ∞        n+1                                 ∞                                           X 1
             11. X(−1)                                    24. X cos(n=2)                                36.             ln(n)
                       nln(n)                                       n2 +4n                                  n=2 (ln(n))
                  n=2                                          n=1
                                                                                                             ∞
                   ∞                                            ∞                                           X√
                  X n n                                        Xn!                                               n        n
             12.      (−1)    2                           25.         2                                 37.     (  2−1)
                             n +25                                 en                                       n=1
                  n=1                                          n=1
                   ∞ n 2                                        ∞ 2                                          ∞
                  X3n                                          Xn +1                                        X√
             13.                                          26.                                           38.     ( n 2 − 1)
                        n!                                            5n
                  n=1                                          n=1                                          n=1