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Problem 320

Show that: a) $\lim _{n \rightarrow \infty}\left[\left\\{n^{4}+n^{3}-1\right\\} /\left\\{\left(n^{2}+2\right)\left(n^{2}-n-1\right)\right\\}\right]=1$ b) $\lim _{n \rightarrow \infty}\left[1+\left(\mathrm{C} / \mathrm{n}^{2}\right)\right]^{\mathrm{n}}=1$, C a constant.

Expert verified

a) As the degrees of the numerator and denominator are both 4, the limit as n approaches infinity is the ratio of the coefficients of the highest-degree terms, which is \(\frac{1}{1} = 1\).
\[
\lim_{n \rightarrow \infty} \left[\frac{n^4+n^3-1}{(n^2+2)(n^2-n-1)}\right] = 1
\]
b) Using L'Hôpital's rule and exponent properties, we find the limit of the power function as n approaches infinity to be a constant:
\[
\lim_{n \rightarrow \infty}\left[1+\left(\frac{C}{n^2}\right)\right]^n = 1
\]

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Chapter 11

Find a) $\lim _{\mathrm{n} \rightarrow \infty}\left[\\{(3 \mathrm{n}) !\\} /\left\\{\mathrm{n}^{3 \mathrm{n}}\right\\}\right]^{1 / \mathrm{n}}$ b) for the sequence of vectors $\left[\\{\mathrm{m} /(\mathrm{m}+1)\\}, 2^{-\mathrm{m}},\\{1+(1 / \mathrm{m})\\}^{\mathrm{m}}\right], \mathrm{m}=1,2, \ldots\( in \)\mathrm{R}^{3}$ the limit as \(\mathrm{m} \rightarrow \infty\)

Chapter 11

For any sequence \(\left\\{a_{n}\right\\}\) of positive number, prove: a) \(\lim _{n \rightarrow \infty}\) Sup $a_{n}{ }^{1 / n} \leq \lim _{n \rightarrow \infty}\( Sup \)\left[\left\\{a_{n+1}\right\\} /\left\\{a_{n}\right\\}\right]$ b) $\lim _{n \rightarrow \infty} \inf \left[\left\\{a_{n+1}\right\\} /\left\\{a_{n}\right\\}\right] \leq \lim _{n \rightarrow \infty}\( inf \)a_{n}{ }^{1 / n}$

Chapter 11

Prove the following theorems: a) Let the functions \(\left\\{\mathrm{f}_{\mathrm{n}}(\mathrm{x})\right\\}\) be defined on the interval \(\mathrm{a} \leq \mathrm{x} \leq \mathrm{b}\) and let \(\mathrm{f}_{n} \rightarrow \mathrm{f}\) uniformly on this interval. Then, if each of the functions \(\mathrm{f}_{\mathrm{n}}\) is continuous at a point \(\mathrm{x}_{0}\), the limit function \(\mathrm{f}\) is also continuous at \(\mathrm{x}_{0}\). In addition, if each \(\mathrm{f}\) is continuous on the entire interval, so is \(\mathrm{f}\). b) Suppose that the functions \(\mathrm{f}_{\mathrm{n}}(\mathrm{x})\) are continuous on the closed interval $\mathrm{a} \leq \mathrm{x} \leq \mathrm{b}$, and suppose that they converge uniformly on this interval to the limit function \(\mathrm{f}(\mathrm{x})\). Then $$ b_{a} f(x) d x=\lim _{n \rightarrow \infty} b_{a} f_{n}(x) d x $$ c) Let the functions \(\mathrm{f}_{\mathrm{n}}(\mathrm{x})\) be defined and have continuous derivatives on the interval $\mathrm{a} \leq \mathrm{x} \leq \mathrm{b}$. If the sequence \(\left\\{\mathrm{f}_{\mathrm{n}}^{\prime}(\mathrm{x})\right\\}\) is uniformly convergent on the interval, and if the sequence \(\left\\{\mathrm{f}_{\mathrm{n}}(\mathrm{x})\right\\}\) is convergent, with limit function \(\mathrm{f}(\mathrm{x})\), then \(\mathrm{f}\) is differentiable and $$ \mathrm{f}(\mathrm{x})=\lim _{\mathrm{n} \rightarrow \infty} \mathrm{f}_{\mathrm{n}}^{\prime}(\mathrm{x}) $$

Chapter 11

Let \(\left\\{p_{n}\right\\}\) sequence in a metric space \(X\). Prove: a) \(\left\\{p_{n}\right\\}\) converges to \(P \in X\) if and only if every neighborhood of \(p\) contains all but finitely many of the terms of \(\left\\{p_{n}\right\\}\). b) If \(p \in X, q \in X\) and if \(\left\\{p_{n}\right\\}\) converges to \(p\) and to \(q\), then \(q=p\) c) If \(\left\\{p_{n}\right\\}\) converges, then \(\left\\{p_{n}\right\\}\) is bounded. d) If \(E \subset X\) and if \(p\) is a limit point of \(E\), then there is a sequence \(\left\\{p_{n}\right\\}\) in \(E\) such that $$ \mathrm{p}=\lim _{\mathrm{n} \rightarrow \infty} \mathrm{p}_{\mathrm{n}} $$

Chapter 11

Evaluate the limit $\lim _{\mathrm{x} \rightarrow \infty}\left[\left\\{\sqrt{(\log \mathrm{x}) \log (\log \mathrm{x})\\}} /\left\\{\mathrm{e}^{\sqrt{\mathrm{x}}}\right\\}\right]\right.$

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