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icefox-saber committed Aug 14, 2024
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2 changes: 1 addition & 1 deletion 404.html
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2 changes: 1 addition & 1 deletion Chap01/Problems/1-1/index.html
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2 changes: 1 addition & 1 deletion Chap02/Problems/2-1/index.html
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2 changes: 1 addition & 1 deletion Chap02/Problems/2-2/index.html
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2 changes: 1 addition & 1 deletion Chap02/Problems/2-4/index.html
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2 changes: 1 addition & 1 deletion Chap03/3.1/index.html
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2 changes: 1 addition & 1 deletion Chap03/3.2/index.html
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2 changes: 1 addition & 1 deletion Chap03/3.3/index.html
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2 changes: 1 addition & 1 deletion Chap03/Problems/3-1/index.html
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2 changes: 1 addition & 1 deletion Chap03/Problems/3-2/index.html
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2 changes: 1 addition & 1 deletion Chap03/Problems/3-3/index.html
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2 changes: 1 addition & 1 deletion Chap03/Problems/3-4/index.html
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2 changes: 1 addition & 1 deletion Chap03/Problems/3-5/index.html
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2 changes: 1 addition & 1 deletion Chap03/Problems/3-6/index.html
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2 changes: 1 addition & 1 deletion Chap03/Problems/3-7/index.html
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2 changes: 1 addition & 1 deletion Chap04/4.1/index.html
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2 changes: 1 addition & 1 deletion Chap04/4.2/index.html
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2 changes: 1 addition & 1 deletion Chap04/4.4/index.html
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2 changes: 1 addition & 1 deletion Chap04/4.6/index.html
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18 changes: 10 additions & 8 deletions Chap04/4.7/index.html
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Expand Up @@ -1666,16 +1666,18 @@ <h2 id="47-5">4.7-5<a class="headerlink" href="#47-5" title="Permanent link">&pa
<p>$$
\begin{aligned}
&amp; \frac{3}{3^p}+\frac{8}{4^p}=1\cr
&amp; \frac{3}{3^1}+\frac{8}{4^1}=3\cr
&amp; \frac{3}{3^{1.5}}+\frac{8}{4^{1.5}}=3^{-(1/2)}+1\cr
&amp; \frac{3}{3^2}+\frac{8}{4^2}=\frac{5}{6}\cr
\implies &amp; 1&lt;p&lt;2\cr
\implies &amp; 1.5&lt;p&lt;2\cr
T(n) &amp; =\Theta(n^p(1+\int_{1}^{n}\frac{f(x)}{x^{p+1}}dx))\cr
&amp; =\Theta(n^p(1+\int_{1}^{n}\frac{x^2/\lg x}{x^{p+1}}dx))\cr
&amp; =\Theta(n^p(1+\int_{1}^{n}\frac{1}{x^{p-1}\ln x}dx))\cr
&amp; = \Omega(n^p(1+\int_{1}^{n}\frac{1}{x\ln x}dx))\cr
&amp; = \Omega(n^p(1+\left[\ln\ln x\right]_1^{n}))\cr
&amp; = \Omega(n^p(1+\ln\ln n-\ln\ln 1))\cr
&amp; = \infty\cr
&amp; =\Theta(n^p(1+\int_{1}^{n}\frac{1}{x^{p-1}\lg x}dx))\cr
&amp; =\Omega(n^p(1+\lg n\int_{1}^{n}\frac{1}{x^{p-1}}dx))\cr
&amp; =\Omega(n^p(1+n^{2-p}\lg n ))\cr
&amp; =\Omega(n^2/\lg n)\cr
T(n) &amp; =O(n^p(1+\lg 1(\text{regarded as constant})\int_{1}^{\sqrt{n}}\frac{1}{x^{p-1}}dx+\lg \sqrt{n}\int_{\sqrt{n}}^{n}\frac{1}{x^{p-1}}dx))\cr
&amp; = O(n^p(1+\Theta(n^{(p-1)/2})+\Theta(n^{p-1}/\lg n)))\cr
&amp; =O(n^2/\lg n)\cr
\end{aligned}
$$</p>
<p><strong>c</strong>.</p>
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2 changes: 1 addition & 1 deletion Chap04/Problems/4-1/index.html
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2 changes: 1 addition & 1 deletion Chap04/Problems/4-2/index.html
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