a. Assuming that limn →∞(1 / n^c)=0 if c is any positive con- limn →∞ (lnn)/(n^c)=0 if c is any

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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 con- $$\lim _{n \rightarrow \infty} \frac{\ln n}{n^{c}}=0$$ 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 $\left|1 / n^{c}-0\right|<\epsilon$ if $n>N ? )$

a. Assuming that $\lim _{n \rightarrow \infty}\left(1 / n^{c}\right)=0$ if $c$ is any positive con-
$$\lim _{n \rightarrow \infty} \frac{\ln n}{n^{c}}=0$$
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 $\left|1 / n^{c}-0\right|<\epsilon$ if $n>N ? )$

Key Concepts

Exponential and Polynomial Growth

Understanding the growth rates of functions is crucial to the problem. In this case, the denominator n^c grows without bound for any positive c as n increases, which intuitively means that its reciprocal, 1/n^c, must shrink towards 0. Recognizing the relationship between polynomial growth (n^c) and its inverse is key to understanding why the limit is 0.

Logarithmic Comparison

The use of logarithms for comparing functions comes in handy for dealing with exponents and determining how large n must be to satisfy a given condition. By taking logarithms, one can solve inequalities that arise from the epsilon-definition, thus providing a method for finding an appropriate index N such that 1/n^c is less than a given ?.

Limit of a Sequence

This concept involves analyzing the behavior of a sequence as its index grows arbitrarily large. When we say a sequence converges to a limit, we mean that the sequence's terms get arbitrarily close to a particular value, called the limit, as the index goes to infinity. In the context of the problem, it establishes that as n increases indefinitely, the value of 1/n^c approaches 0 for any positive constant c.

Epsilon-Definition of Limits

The epsilon-definition provides a precise way to define and prove that a sequence converges to a limit. It states that for every positive number ? (no matter how small), there exists an index N such that for all n > N, the absolute difference between the sequence term and the limit is less than ?. This definition is central to the example given, where one must determine an appropriate N to ensure the sequence term is within a chosen ? of 0.

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