209 lines
9.3 KiB
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209 lines
9.3 KiB
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<h2>Convergents of e</h2><div class="info" style="cursor:help;width:200px;margin-bottom:10px;"><h3>Problem 65</h3><span style="width:300px;color:#666;">Published on Friday, 12th March 2004, 06:00 pm; Solved by 13567</span></div>
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<div class="problem_content" role="problem">
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<p>The square root of 2 can be written as an infinite continued fraction.</p>
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<div style="margin-left:20px;">
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<table border="0" cellspacing="0" cellpadding="0">
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<tr>
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<td><img src='images/symbol_radic.gif' width='14' height='16' alt='√' border='0' style='vertical-align:middle;' />2 = 1 +</td>
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<td colspan="4"><div style="text-align:center;">1<br /><img src="images/blackdot.gif" width="135" height="1" alt="" /><br /></div></td>
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</tr>
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<tr>
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<td> </td>
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<td>2 +</td>
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<td colspan="3"><div style="text-align:center;">1<br /><img src="images/blackdot.gif" width="110" height="1" alt="" /><br /></div></td>
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</tr>
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<tr>
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<td> </td>
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<td> </td>
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<td>2 +</td>
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<td colspan="2"><div style="text-align:center;">1<br /><img src="images/blackdot.gif" width="85" height="1" alt="" /><br /></div></td>
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</tr>
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<tr>
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<td> </td>
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<td> </td>
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<td> </td>
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<td>2 +</td>
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<td><div style="text-align:center;">1<br /><img src="images/blackdot.gif" width="60" height="1" alt="" /><br /></div></td>
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</tr>
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<tr>
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<td> </td>
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<td> </td>
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<td> </td>
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<td> </td>
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<td>2 + ...</td>
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</tr>
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</table>
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</div>
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<p>The infinite continued fraction can be written, <img src='images/symbol_radic.gif' width='14' height='16' alt='√' border='0' style='vertical-align:middle;' />2 = [1;(2)], (2) indicates that 2 repeats <i>ad infinitum</i>. In a similar way, <img src='images/symbol_radic.gif' width='14' height='16' alt='√' border='0' style='vertical-align:middle;' />23 = [4;(1,3,1,8)].</p>
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<p>It turns out that the sequence of partial values of continued fractions for square roots provide the best rational approximations. Let us consider the convergents for <img src='images/symbol_radic.gif' width='14' height='16' alt='√' border='0' style='vertical-align:middle;' />2.</p>
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<div style="margin-left:20px;">
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<table border="0" cellspacing="0" cellpadding="0">
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<tr>
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<td>1 +</td>
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<td><div style="text-align:center;">1<br /><img src="images/blackdot.gif" width="15" height="1" alt="" /><br /></div></td>
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<td>= 3/2</td>
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</tr>
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<tr>
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<td> </td>
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<td><div style="text-align:center;">2</div></td>
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<td> </td>
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</tr>
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</table>
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<table border="0" cellspacing="0" cellpadding="0">
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<tr>
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<td>1 +</td>
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<td colspan="2"><div style="text-align:center;">1<br /><img src="images/blackdot.gif" width="50" height="1" alt="" /><br /></div></td>
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<td>= 7/5</td>
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</tr>
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<tr>
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<td> </td>
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<td>2 +</td>
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<td><div style="text-align:center;">1<br /><img src="images/blackdot.gif" width="15" height="1" alt="" /><br /></div></td>
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</tr>
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<tr>
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<td> </td>
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<td> </td>
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<td><div style="text-align:center;">2</div></td>
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<td> </td>
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</tr>
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</table>
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<table border="0" cellspacing="0" cellpadding="0">
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<tr>
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<td>1 +</td>
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<td colspan="3"><div style="text-align:center;">1<br /><img src="images/blackdot.gif" width="80" height="1" alt="" /><br /></div></td>
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<td>= 17/12</td>
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</tr>
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<tr>
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<td> </td>
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<td>2 +</td>
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<td colspan="2"><div style="text-align:center;">1<br /><img src="images/blackdot.gif" width="50" height="1" alt="" /><br /></div></td>
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<td> </td>
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</tr>
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<tr>
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<td> </td>
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<td> </td>
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<td>2 +</td>
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<td><div style="text-align:center;">1<br /><img src="images/blackdot.gif" width="15" height="1" alt="" /><br /></div></td>
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<td> </td>
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</tr>
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<tr>
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<td> </td>
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<td> </td>
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<td> </td>
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<td><div style="text-align:center;">2</div></td>
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<td> </td>
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</tr>
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</table>
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<table border="0" cellspacing="0" cellpadding="0">
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<tr>
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<td>1 +</td>
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<td colspan="4"><div style="text-align:center;">1<br /><img src="images/blackdot.gif" width="110" height="1" alt="" /><br /></div></td>
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<td>= 41/29</td>
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</tr>
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<tr>
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<td> </td>
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<td>2 +</td>
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<td colspan="3"><div style="text-align:center;">1<br /><img src="images/blackdot.gif" width="80" height="1" alt="" /><br /></div></td>
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</tr>
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<tr>
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<td> </td>
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<td> </td>
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<td>2 +</td>
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<td colspan="2"><div style="text-align:center;">1<br /><img src="images/blackdot.gif" width="50" height="1" alt="" /><br /></div></td>
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<td> </td>
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</tr>
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<tr>
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<td> </td>
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<td> </td>
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<td> </td>
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<td>2 +</td>
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<td><div style="text-align:center;">1<br /><img src="images/blackdot.gif" width="15" height="1" alt="" /><br /></div></td>
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<td> </td>
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</tr>
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<tr>
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<td> </td>
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<td> </td>
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<td> </td>
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<td> </td>
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<td><div style="text-align:center;">2</div></td>
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<td> </td>
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</tr>
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</table>
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</div>
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<p>Hence the sequence of the first ten convergents for <img src='images/symbol_radic.gif' width='14' height='16' alt='√' border='0' style='vertical-align:middle;' />2 are:</p>
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<div class="info">1, 3/2, 7/5, 17/12, 41/29, 99/70, 239/169, 577/408, 1393/985, 3363/2378, ...</div>
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<p>What is most surprising is that the important mathematical constant,<br /><i>e</i> = [2; 1,2,1, 1,4,1, 1,6,1 , ... , 1,2<i>k</i>,1, ...].</p>
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<p>The first ten terms in the sequence of convergents for <i>e</i> are:</p>
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<div class="info">2, 3, 8/3, 11/4, 19/7, 87/32, 106/39, 193/71, 1264/465, 1457/536, ...</div>
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<p>The sum of digits in the numerator of the 10<sup>th</sup> convergent is 1+4+5+7=17.</p>
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<p>Find the sum of digits in the numerator of the 100<sup>th</sup> convergent of the continued fraction for <i>e</i>.</p>
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