How do you use the integral test to find whether the following series converges or diverges #sum( 1/(n*ln(n)^0.5) )#?

1 Answer
Ultrilliam
Apr 15, 2018

Does not converge

Explanation:

The function # 1/(x*sqrt( ln(x))# is positive, continuous, monotone decreasing over the interval #[N to oo)#, #N in mathbf Z^+# and so # sum _{n=N}^{oo } \ 1/(n*sqrt( ln(n)) # converges to a real number iff the following improper integral is finite:

  • #int_N^(oo) 1/(x*sqrt( ln(x)) \ dx#

#int_N^(oo) 1/(x*sqrt( ln(x)) \ dx#

#= 2 int_N^(oo) d(sqrt (ln(x))) #

#= 2 [ \ sqrt (ln(x)) \ ]_N^(oo)#

#= 2 ( sqrt (ln(omega)) |_(omega to oo) - sqrt (ln(N)\ ) \ )#

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