Determining the Convergence of the Series Σn2∞ 1/ [ln^n n]

The problem at hand involves determining the convergence of a series:

$$sum_{n2}^{infty} frac{1}{(ln n)^n}$$

We will utilize the Ratio Test and the Cauchy Condensation Test to establish the convergence of this series.

Applying the Ratio Test

The Ratio Test states that for a series $$sum a_n$$, if we compute the limit:

$$L lim_{n to infty} left| frac{a_{n 1}}{a_n} right|$$

We can use this test to determine the convergence of the series:

Step-by-Step Application of the Ratio Test

Let us define:

$$a_n frac{1}{(ln n)^n}$$

We need to compute the term:

$$a_{n 1} frac{1}{(ln (n 1))^{n 1}}$$

The ratio $$frac{a_{n 1}}{a_n}$$ can be computed as:

$$frac{a_{n 1}}{a_n} frac{frac{1}{(ln (n 1))^{n 1}}}{frac{1}{(ln n)^n}} frac{(ln n)^n}{(ln (n 1))^{n 1}}$$

This simplifies to:

$$frac{a_{n 1}}{a_n} frac{(ln n)^n}{(ln (n 1))^{n 1}} frac{(ln n)^n}{(ln (n 1))^n (ln (n 1))} frac{(ln n)^n}{(ln n ln (1 frac{1}{n}))^n (ln (n 1))}$$

Approximating at large n:

$$ln (n 1) sim ln n$$

Thus, we have:

$$lim_{n to infty} frac{a_{n 1}}{a_n} lim_{n to infty} frac{(ln n)^n}{(ln n)^n (ln (n 1))} lim_{n to infty} frac{1}{ln (n 1)} 0$$

Since $$L 0 , the series $$sum_{n2}^{infty} frac{1}{(ln n)^n}$$ converges by the Ratio Test.

Applying the Cauchy Condensation Test

To further confirm convergence, we apply the Cauchy Condensation Test:

Consider the series:

$$S_n sum_{k2}^{n-1} frac{1}{(ln k)^k}$$

Applying the test, we obtain:

$$frac{2^k}{(ln 2^k)^{2^k}} frac{2^k}{k^{2^k} (ln 2)^{2^k}}$$

This can be simplified to:

$$frac{2^k}{k^{2^k} (ln 2)^{2^k}} frac{2^k}{k^{2^k}} cdot frac{1}{(ln 2)^{2^k}}$$

Next, we consider the ratio for the next term:

$$frac{2^{k-1}}{2^k} cdot frac{(k 1)^{2^{k-1}}}{k^{2^k}} cdot frac{(ln 2)^{2^k}}{(ln 2)^{2^{k-1}}} frac{1}{2} cdot frac{(k 1)^{2^{k-1}}}{k^{2^k}} cdot (ln 2)^{2^k} / (ln 2)^{2^{k-1}}$$

This further simplifies to:

$$frac{1}{2} cdot frac{(k 1)^{2^{k-1}}}{k^{2^k}} cdot (ln 2)^{2^{k-1}} frac{1}{2} cdot frac{(k 1)^{2^{k-1}}}{k^{2^k}} cdot (ln 2)^{2^{k-1}} o(1) text{ as } k to infty$$

Since this ratio approaches 0 as $$k to infty$$, the series converges by the Cauchy Condensation Test.

Conclusion

By both the Ratio Test and the Cauchy Condensation Test, we can conclude that the series $$sum_{n2}^{infty} frac{1}{(ln n)^n}$$ converges.

For further reference, proofs of these tests are available on the web.