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Understanding limits of sequences helps determine whether a sequence converges or diverges. A sequence converges if the terms become closer and closer to a specific value as the sequence progresses, which is known as the limit. Conversely, a sequence diverges if it does not approach any particular value.
As seen in the step-by-step solution, when we examine the sequence \(a_n = \frac{\ln n}{n^{1/n}}\), we must consider how both the numerator and the denominator behave as \( n \to \infty \).
- The numerator \( \ln n \) increases without an upper bound as \( n \) grows.- The denominator \( n^{1/n} \) tends towards 1.
Since \( \ln n \) grows much faster than the minor changes in \( n^{1/n} \), the overall behavior of \( \frac{\ln n}{n^{1/n}} \) is to grow more significant, leading to the conclusion that the sequence diverges.

Logarithms describe a powerful mathematical tool used to solve various problems involving exponential growth or decay. It's a function that determines the power to which a base, often e (the natural logarithm), is raised to obtain a particular number.
For the sequence \(a_n = \frac{\ln n}{n^{1/n}}\), the numerator \( \ln n \) plays a crucial role in the sequence's behavior.
### Important Properties to Remember

• The natural logarithm \( \ln x \) grows slower than any positive power \( x^c \) for \( c > 0 \) but faster than any logarithm with another base.
• As \( n \to \infty \), \( \ln n \) also \( \to \infty \). This ensures any sequence or function involving \( \ln n \) will ultimately depend significantly on this unbounded growth.

This property clarifies why, in our exercise, the term \( \ln n \) dictates the sequence's ultimate behavior, leading to its divergence.

Asymptotic behavior involves looking at how functions or sequences behave as they head towards infinity or another significant point. It's about understanding the "end behavior" of mathematical entities.
For the sequence in question, \(a_n = \frac{\ln n}{n^{1/n}}\), the asymptotic behavior is pivotal in understanding whether the sequence will converge to a finite limit or diverge.
### Key Aspects

• As \( n \to \infty \), \( n^{1/n} \to 1 \). This means it ultimately behaves almost like dividing by 1, barely impacting the numerator
  compared to infinite growth.
• Consequently, with \( \ln n \) increasing substantially more whenever \( n \) approaches larger values, \( \ln n \) dominantly decides the ratio's behavior.

This asymptotic insight explains why \( a_n \) cannot stabilize or approach a fixed value, ultimately characterizing it as a divergent sequence.