Introduction

Understanding the convergence of infinite series is a fundamental concept in mathematical analysis. This article focuses on demonstrating the convergence of the series (sum_{n2}^{infty} frac{1}{ln n^n}). We will explore the use of the root test and the squeeze theorem to establish the convergence of this series.

Root Test Application

The root test is a powerful tool for determining the convergence of a series of the form (sum a_n). The root test states that if (L limsup_{n toinfty} sqrt[n]{a_n}), then:

if (L 1), the series converges absolutely, if (L 1), the series diverges, if (L 1), the test is inconclusive.

Applying the Root Test

For our series (sum_{n2}^{infty} frac{1}{ln n^n}), we have (a_n frac{1}{ln n^n}).

To apply the root test, we calculate:

[sqrt[n]{a_n} sqrt[n]{frac{1}{ln n^n}} frac{1}{ln n}]

Next, we find the limit as (n to infty):

[lim_{n to infty} sqrt[n]{a_n} lim_{n to infty} frac{1}{ln n} 0]

Since (0 1), the root test confirms that the series converges.

Squeeze Theorem Application

To further solidify our understanding, we can use the squeeze theorem. Consider the inequality:

[frac{1}{ln n} geq frac{1}{2}] for all (n geq 8).

This implies:

[frac{1}{ln n^n} leq frac{1}{2^n}] for all (n geq 8).

Using the squeeze theorem, we know that if:

[a_n leq b_n leq c_n] and (sum c_n) converges, then (sum b_n) also converges.

In this case, we have:

[sum_{n2}^{infty} frac{1}{ln n^n} leq sum_{n8}^{infty} frac{1}{2^n}]

Since the geometric series (sum_{n8}^{infty} frac{1}{2^n}) converges (it is equal to a finite number), the original series also converges by the squeeze theorem.

Alternative Convergence Tests

Another way to verify convergence is by using the ratio test, which is particularly useful for series involving factorials or exponentials. For our series, we have:

[lim_{n to infty} frac{a_{n 1}}{a_n} lim_{n to infty} frac{frac{1}{ln (n 1)^{n 1}}}{frac{1}{ln n^n}} frac{ln n^n}{ln (n 1)^{n 1}} frac{n cdot ln n}{(n 1) cdot ln (n 1)}]

As (n to infty), we find that:

[lim_{n to infty} frac{n cdot ln n}{(n 1) cdot ln (n 1)} 1]

Using the d'Alembert criterion (ratio test), if the limit is 1, the test is inconclusive. However, as we have seen in the root test, the series still converges.

Conclusion

Through the application of the root test and the squeeze theorem, we have demonstrated that the series (sum_{n2}^{infty} frac{1}{ln n^n}) converges. This method provides a comprehensive and rigorous approach to verifying the convergence of infinite series involving logarithmic functions.

For a deeper understanding of series convergence and advanced tests, refer to the following resources:

Root Test on Wikipedia Squeeze Theorem on Wikipedia Ratio Test on Wikipedia

By exploring these methods, you can gain a strong foundation in the analysis of series and their convergence.