Does the series converge or diverge?

Question

Converge or Diverge?
#oo#
#sum_(n=1)n/(n^2+2n+3)#

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Answer 1 · 2017-11-02T15:47:48

The series

#sum_(n=1)^oo n/(n^2+2n+3)#

is divergent.

Use the limit comparison test with the harmonic series.

As we know that:

#sum_(n=1)^oo 1/n#

is divergent, and:

#lim_(n->oo) (n/(n^2+2n+3))/(1/n) = lim_(n->oo) n^2/(n^2+2n+3) = 1#

is finite, then the two series have the same character, which means that also:

#sum_(n=1)^oo n/(n^2+2n+3)#

is divergent.