Question

Use the Ratio Test or the Root Test to determine the convergence or divergence of the series. $$ \frac{1}{(\ln 3)^{3}}+\frac{1}{(\ln 4)^{4}}+\frac{1}{(\ln 5)^{5}}+\frac{1}{(\ln 6)^{6}}+\cdots $$

Short answer

Based on the Ratio Test, the given series is absolutely convergent.

Step-by-step solution

Identify the nth term of the series

First, identify the nth term of the series. It is given by: \(a_n = \frac{1}{(\ln n)^{n}}\)

Calculate the ratio of (n+1)th term to nth term

Now calculate the ratio of the \((n+1)\)th term to the nth term. The ratio is given by: \(r_n = \frac{a_{n+1}}{a_n} = \frac{(\ln n)^n}{(\ln (n+1))^{n+1}}\)

Calculate the Limit

Now, calculate the limit of \(r_n\) as \(n\) approaches infinity. As \(n\) becomes very large, \(\ln (n+1)\) will exceed \(\ln n\), so the ratio \(r_n\) will go to zero. Thus,\(\lim_{{n}\to\infty} {r_n} = 0\)

Apply the Ratio Test

The Ratio Test states that if the limit obtained in step 3 is less than 1 (which it is in this case, 0<1), then the series is absolutely convergent.