Root Test for for Convergence of an Infinite Series

Key Questions

  • How do you know when to use the Root Test for convergence of a series?

    I would use Root Test when the terms of the series are in the form of some expression to the nth power; otherwise, I would try other tests first.

    Example

    Let us look at examine the convergence of the series:

    #sum_{n=1}^infty({2n}/{5-3n})^n#

    By Root Test,

    #lim_{n to infty}root{n}{|({2n}/{5-3n})^n|}=lim_{n to infty}|{2n}/{5-3n}|#

    by dividing the numerator and the denominator by #n#,

    #=lim_{n to infty}|{2}/{5/n-3}|=|{2}/{0-3}|=2/3<1#

    Hence, the series is absolutely convergent.

    I hope that this was helpful.

    Wataru · 1 · Nov 25 2014
  • How do you use the Root Test on the series #sum_(n=1)^oo((n^2+1)/(2n^2+1))^(n)# ?

    Let #a_n=({n^2+1]/{2n^2+1})^n#.

    By Root Test,

    #lim_{n to infty}root[n]{|a_n|}=lim_{n to infty}root[n]{|({n^2+1}/{2n^2+1})^n|}#

    by cancelling out the nth-root and the nth-power,

    #=lim_{n to infty}{n^2+1}/{2n^2+1}#

    (Note: the absolute value is not necessary since it is already positive.)

    by dividing by #n^2#,

    #=lim_{n to infty}{1+1/n^2}/{2+1/n^2}={1+0}/{2+0}=1/2<1#

    Hence, the series converges.

    I hope that this was helpful.

    Wataru · · Oct 16 2014
  • What is the Root Test for Convergence of an Infinite Series?

    Root Test

    If #lim_{n to infty}root[n]{|a_n|}<1#, then #sum_{n=1}^inftya_n# converges.
    If #lim_{n to infty}root[n]{|a_n|}>1#, then #sum_{n=1}^inftya_n# diverges.
    If #lim_{n to infty}root[n]{|a_n|}=1#, then it is inconclusive.

    I hope that this was helpful.

    Wataru · · Oct 17 2014

Questions

Tests of Convergence / Divergence