Series convergence help

1. 

inf

∑   2 / [ n ( n + 1) ]     converges to  2   

1

This series converges by a comparison test  to  2 / n^2 [ which conveges by the integral test ]

2 / n^2                       2       1/2     2/9  ..........      

2 / [ n ( n + 1) ]          1       1/3     1/6  ..........

Both series  >  0  for all terms....and, term by term, the second series is smaller than the second

So....if 2/n^2 converges....so does  2 / [ n ( n + 1) ] 

2.

inf

∑   [ n / ( n^2 -  1) ]    

2

This series diverges by a comparison test  to 1 / n  [ which diverges by the p - series test]

1/n                         1/2        1/3      1/4  .........

n / ( n^2 -  1)         2/3        3/8       4/15  ........

Both series  >  0  for all terms....and, term by term, the first series is smaller than the second

So....if 1/n diverges....so does  n / ( n^2 -  1)  

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