1.
Show that the limit of a convergent sequence of non-negative numbers is non-negative.
Exercises 5.7 Exercises
2.
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For \(c \gt 1\text{,}\) show that the sequence \((\sqrt[n]{c})\) is convergent by showing that it is decreasing and bounded below by 1.
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Show that \(\sqrt[n]{c} \to 1\text{.}\)
Formulate and prove similar statements for \(0 \lt c \lt 1\) (Hint: consider \(1/c\)).
3.
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Show that the function \(\ln(x)/x\) is decreasing for \(x \ge e\text{.}\)
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Deduce that the sequence \((n^{1/n})\) is decreasing for \(n \ge 3\text{,}\) i.e.\begin{equation*} \sqrt[3]{3} \ge \sqrt[4]{4} \ge \sqrt[5]{5} \ge \cdots\text{.} \end{equation*}
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Conclude that the sequence \((n^{1/n})_{n \ge 1}\) is convergent. (The limit is \(1\) see Example 5.23)
4.
Let \(p(x)\) be a polynomial with positive leading coefficient. Show that \((p(n))^{1/n} \to 1\) as \(n \to \infty\text{.}\)
Hint.
Suppose \(ax^d\) is the leading term of \(p(x)\text{.}\) Then \(0 \lt p(n) \lt 2an^d\) for all sufficiently large \(n\text{.}\) Now apply Example 5.23.
5.
Show that \((n!)^2 \ge n^n\text{.}\) Deduce that \((n!)^{1/n} \to +\infty.\)
6.
Show that \((\sin(n))\) is a divergent sequence.
Hint.
There are a number of ways of show this. Here is an elementary way: Suppose on the contrary that \(\sin(n) \to L\text{.}\) Then both \(\sin(n+1)\) and \(\sin(n-1)\) converge to \(L\) as well. Then use the identities:
\begin{equation*}
\begin{aligned}
\sin(n+1) + \sin(n-1) \equiv 2\sin(n)\cos(1) \\
\sin(n+1) - \sin(n-1) \equiv 2\sin(1)\cos(n)
\end{aligned}
\end{equation*}
to conclude that \(\sin(n) \to 0\) and that \(\cos(n)\to 0\) as well. This gives us a contradiction as \(\sin^2(n) + \cos^2(n) \equiv 1\text{.}\)
7.
Show the divergence of the following series.
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\(\displaystyle \sum \sin(n)\)
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\(\displaystyle \sum \frac{n}{n+1}\)
8.
Compute the following sum using the technique in Example 5.71.
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\(\ds \sum_{n=0}^{\infty} \frac{n^2-n}{3^n}\) Hint: differentiate the relation \(\frac{1}{1-x} = \sum_{n=0}^{\infty} x^n\) twice.
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\(\ds \sum_{n=0}^{\infty} \frac{n^2}{3^n}\) Hint: it is the sum of the series in Part (a) and \(\ds \sum_{n=0}^{\infty} \frac{n}{3^n}\text{.}\)
