Show that the root test fails when applied to the series: sum (k=2 -> inf) 1/k * (1/ln k)^(3/2).

Show that the root test fails when applied to the series:

sum (k=2 -> inf) (1/k)*(1/ln k)^(3/2).

Then we have to show the convergence of this series using any other method. Any recommendations?

Using the root test:
r = lim(k→∞) |(1/k)*(1/ln k)^(3/2)|^(1/k)
..= lim(k→∞) (1/k^(1/k)) * (1/(ln k)^(1/k))^(3/2).
..= (1/1) * (1/1)^(3/2)
..= 1, because:

(a) L = lim(k→∞) k^(1/k)
==> ln L = lim(k→∞) ln(k)/k = lim(k→∞) (1/k)/1 = 0
==> L = e^0 = 1, and

(b) L = lim(k→∞) (ln k)^(1/k)
==> ln L = lim(k→∞) ln(ln k)/k = lim(k→∞) (1/(k ln k)/1 = 0
==> L = e^0 = 1.

Since r = 1, the ratio test is inconclusive.
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However, we can use the integral test.
∫(x = 2 to ∞) dx / [x * (ln x)^(3/2)]
= ∫(w = ln 2 to ∞) dw/w^(3/2), letting w = ln x
= ∫(w = ln 2 to ∞) w^(-3/2) dw
= -2w^(-1/2) {for w = ln 2 to ∞}
= 2 (ln 2)^(1/2).

Since the integral converges, so does the original series.

I hope this helps!