Question

Show that the Ratio Test and the Root Test are both inconclusive for the logarithmic \(p\) -series \(\sum_{n=2}^{\infty} \frac{1}{n(\ln n)^{p}}\)

Short answer

Both the Ratio Test and the Root Test are inconclusive for the logarithmic \(p\)-series \(\sum_{n=2}^{\infty} \frac{1}{n(\ln n)^{p}}\) since they both result in a limit of 1.

Step-by-step solution

Applying the Ratio Test

For the Ratio Test, we take the limit as \(n\) goes to infinity of the absolute value of the ratio between the \((n+1)\)th term and the \(n\)th term of the series. For our series, this means we need to solve \(\lim_{{n \to \infty}} \left| \frac{n(\ln n)^{p}}{(n+1)(\ln(n+1))^{p}} \right|\). As you simplify, you'll see that the limit equals 1, which is inconclusive for the Ratio Test (we need the limit to be less than 1 for convergence and greater than 1 for divergence).

Applying the Root Test

For the Root Test, we calculate the limit as \(n\) tends to infinity of the \(n\)th root of the absolute value of the \(n\)th term of the series. For our given series, we compute \(\lim_{{n \to \infty}} \left( \frac{1} {n(\ln n)^{p}} \right)^{1/n}\). As you simplify, you'll end up with 1 which once again is inconclusive for the test. The Root Test only determines a series converges if the limit is less than 1, and diverges if it is greater than 1.

Conclusion

With both tests returning a limit of 1, neither test is able to decisively determine whether our series converges or diverges. Therefore, the Ratio Test and the Root Test are both inconclusive for our logarithmic \(p\)-series.