a. Assuming that limn →∞(1 / n^c)=0 if c is any positive constant, show that limn →∞ (lnn)/(n^t)

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A. Assuming that limn →∞(1 / n^c)=0 if c is any positive...

a. Assuming that $\lim _{n \rightarrow \infty}\left(1 / n^{c}\right)=0$ if $c$ is any positive constant, show that $$\lim _{n \rightarrow \infty} \frac{\ln n}{n^{t}}=0$$ if $c$ is any positive constant. b. Prove that $\lim _{n \rightarrow \infty}\left(1 / n^{c}\right)=0$ if $c$ is any positive constant. (Hint: If $\epsilon=0.001$ and $c=0.04,$ how large should $N$ be to ensure that $\left.\left|1 / n^{c}-0\right|<\epsilon \text { if } n>N ?\right)$

a. Assuming that $\lim _{n \rightarrow \infty}\left(1 / n^{c}\right)=0$ if $c$ is any positive constant, show that
$$\lim _{n \rightarrow \infty} \frac{\ln n}{n^{t}}=0$$ if $c$ is any positive constant.
b. Prove that $\lim _{n \rightarrow \infty}\left(1 / n^{c}\right)=0$ if $c$ is any positive constant. (Hint: If $\epsilon=0.001$ and $c=0.04,$ how large should $N$ be to ensure that $\left.\left|1 / n^{c}-0\right|<\epsilon \text { if } n>N ?\right)$

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Key Concepts

Epsilon-N Definition of a Limit

The epsilon-N definition provides a formal framework for proving convergence of sequences. To show that a sequence converges to a limit L, one must demonstrate that for every positive number ?, there exists an integer N such that for all n > N, the absolute difference between the n-th term and L is less than ?. This method is often used to rigorously establish limits, including showing that sequences like 1/n^c tend to 0 as n approaches infinity when c is a positive constant.

Asymptotic Growth Comparison

This concept involves comparing the rates at which different functions increase as their variables approach infinity. In many problems, it is essential to recognize that logarithmic functions, such as ln(n), grow much more slowly than any positive power of n. This understanding is fundamental in proving that when a slowly growing numerator is coupled with a rapidly growing denominator, the resulting fraction tends to zero.

Limits of Sequences

Limits of sequences is a core concept in calculus and analysis that deals with the behavior of sequences as their index tends to infinity. In this subject, determining whether a sequence converges (in this case, converging to 0) is crucial. It involves evaluating whether the terms of the sequence get arbitrarily close to a specific value beyond a certain index.

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