document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209919>Question 1209919</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let x_1, x_2, \dots, x_n be real numbers.  If<BR>\n" );
document.write( "x_1^2 + 2x_2^2 + \dots + nx_n^2 = 1,<BR>\n" );
document.write( "then find the maximum value of (x_1 + x_2/2 + \dots + x_n/n)^2, in terms of n. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #851057 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=851057\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let $a_k = x_k \sqrt{k}$ for $k = 1, 2, \dots, n$.<BR>\n" );
document.write( "Then the given condition becomes<BR>\n" );
document.write( "$$\sum_{k=1}^n x_k^2 k = \sum_{k=1}^n a_k^2 = 1$$\r<BR>\n" );
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document.write( "We want to find the maximum value of<BR>\n" );
document.write( "$$S = \left( \sum_{k=1}^n \frac{x_k}{k} \right)^2 = \left( \sum_{k=1}^n \frac{a_k}{k\sqrt{k}} \right)^2$$\r<BR>\n" );
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document.write( "By the Cauchy-Schwarz inequality, we have<BR>\n" );
document.write( "$$\left( \sum_{k=1}^n \frac{a_k}{k\sqrt{k}} \right)^2 \le \left( \sum_{k=1}^n a_k^2 \right) \left( \sum_{k=1}^n \frac{1}{k^3} \right)$$<BR>\n" );
document.write( "Since $\sum_{k=1}^n a_k^2 = 1$, we have<BR>\n" );
document.write( "$$S \le \sum_{k=1}^n \frac{1}{k^3}$$\r<BR>\n" );
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document.write( "The maximum value of $\left( \sum_{k=1}^n \frac{x_k}{k} \right)^2$ is $\sum_{k=1}^n \frac{1}{k^3}$.\r<BR>\n" );
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document.write( "To achieve equality in Cauchy-Schwarz inequality, we need<BR>\n" );
document.write( "$$\frac{a_1}{1\sqrt{1}} = \frac{a_2}{2\sqrt{2}} = \dots = \frac{a_n}{n\sqrt{n}} = c$$<BR>\n" );
document.write( "where $c$ is a constant.<BR>\n" );
document.write( "Then $a_k = ck\sqrt{k}$, so $x_k \sqrt{k} = ck\sqrt{k}$, which means $x_k = ck$.<BR>\n" );
document.write( "Substituting into the given condition, we have<BR>\n" );
document.write( "$$\sum_{k=1}^n kx_k^2 = \sum_{k=1}^n k(ck)^2 = c^2 \sum_{k=1}^n k^3 = 1$$<BR>\n" );
document.write( "$$c^2 = \frac{1}{\sum_{k=1}^n k^3}$$<BR>\n" );
document.write( "$$c = \frac{1}{\sqrt{\sum_{k=1}^n k^3}}$$<BR>\n" );
document.write( "Then $x_k = \frac{k}{\sqrt{\sum_{k=1}^n k^3}}$.\r<BR>\n" );
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document.write( "Therefore, the maximum value of $\left( \sum_{k=1}^n \frac{x_k}{k} \right)^2$ is $\sum_{k=1}^n \frac{1}{k^3}$.\r<BR>\n" );
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document.write( "Final Answer: The final answer is $\boxed{\sum_{k=1}^n \frac{1}{k^3}}$<BR>\n" );
document.write( " </SPAN>\n" );
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