Nth Term Test for Divergence of an Infinite Series

Key Questions

  • How do you use the Nth term test on the infinite series #sum_(n=1)^ooln((2n+1)/(n+1))# ?

    By the nth term test (Divergence Test), we can conclude that the posted series diverges.

    Recall: Divergence Test
    If #lim_{n to infty}a_n ne 0#, then #sum_{n=1}^{infty}a_n# diverges.

    Let us evaluate the limit.
    #lim_{n to infty}ln({2n+1}/{n+1})#
    by squeezing the limit inside the log,
    #=ln(lim_{n to infty}{2n+1}/{n+1})#
    by dividing the numerator and the denominator by #n#,
    #=ln(lim_{n to infty}{2n+1}/{n+1}cdot{1/n}/{1/n}) =ln(lim_{n to infty}{2+1/n}/{1+1/n})#
    since #1/n to 0#, we have
    #=ln2ne 0#

    By Divergence Test, we may conclude that
    #sum_{n=1}^{infty}ln({2n+1}/{n+1})# diverges.

    Caution: This test does not detect all divergent series; for example, the harmonic series #sum_{n=1}^{infty}1/n# diverges even though #lim_{n to infty}1/n=0#.

    Wataru · · Sep 3 2014
  • What is Nth Term Test for Divergence of an infinite series?

    Nth Term Test (also called Divergence Test)

    If #lim_{n to infty}|a_n| ne 0#, then #sum_{n=1}^inftya_n# diverges.

    Wataru · · Sep 25 2014

Questions

Tests of Convergence / Divergence