235 lines
8.7 KiB
HTML
235 lines
8.7 KiB
HTML
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<h2>A Kempner-like series</h2><div class="info" style="cursor:help;width:200px;margin-bottom:10px;"><h3>Problem 368</h3><span style="width:300px;color:#666;">Published on Sunday, 22nd January 2012, 01:00 am; Solved by 328</span></div>
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<div class="problem_content" role="problem">
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<P>
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<table ><tr>
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<td>The <b> harmonic series </b> 1</td>
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<td>+</td><td><table class="frac">
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<tr><td>1</td></tr>
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<tr><td class="overline">2</td></tr>
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</table>
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</td>
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<td>+</td>
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<td><table class="frac">
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<tr><td>1</td></tr>
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<tr><td class="overline">3</td></tr>
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</table>
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</td>
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<td>+</td>
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<td><table class="frac">
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<tr><td>1</td></tr>
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<tr><td class="overline">4</td></tr>
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</table>
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</td>
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<td>+ ... is well known to be divergent. </td>
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</tr></table>
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</P>
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<p>
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If we however omit from this series every term where the denominator has a 9 in it, the series remarkably enough converges to approximately 22.9206766193.<BR />
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This modified harmonic series is called the <b>Kempner</b> series.
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</p>
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<P>
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Let us now consider another modified harmonic series by omitting from the harmonic series every term where the denominator has 3 or more equal consecutive digits.
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One can verify that out of the first 1200 terms of the harmonic series, only 20 terms will be omitted.<BR />
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These 20 omitted terms are:
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</P>
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<table>
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<td><table class="frac" style="text-align:center;">
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<tr><td >1</td></tr>
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<tr><td class="overline">111</td></tr>
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</table>
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</td>
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<td>,</td>
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<td><table class="frac" style="text-align:center;">
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<tr><td>1</td></tr>
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<tr><td class="overline">222</td></tr>
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</table>
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</td>
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<td>,</td>
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<td><table class="frac"style="text-align:center;">
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<tr><td>1</td></tr>
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<tr><td class="overline">333</td></tr>
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</table>
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</td>
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<td>,</td>
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<td><table class="frac" style="text-align:center;">
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<tr><td>1</td></tr>
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<tr><td class="overline">444</td></tr>
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</table>
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</td>
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<td>,</td>
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<td><table class="frac" style="text-align:center;">
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<tr><td>1</td></tr>
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<tr><td class="overline">555</td></tr>
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</table>
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</td>
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<td>,</td>
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<td><table class="frac" style="text-align:center;">
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<tr><td>1</td></tr>
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<tr><td class="overline">666</td></tr>
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</table>
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</td>
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<td>,</td>
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<td><table class="frac" style="text-align:center;">
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<tr><td>1</td></tr>
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<tr><td class="overline">777</td></tr>
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</table>
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</td>
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<td>,</td>
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<td><table class="frac" style="text-align:center;">
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<tr><td>1</td></tr>
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<tr><td class="overline">888</td></tr>
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</table>
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</td>
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<td>,</td>
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<td><table class="frac" style="text-align:center;">
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<tr><td>1</td></tr>
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<tr><td class="overline">999</td></tr>
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</table>
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</td>
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<td>,</td>
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<td><table class="frac" style="text-align:center;">
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<tr><td>1</td></tr>
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<tr><td class="overline">1000</td></tr>
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</table>
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</td>
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<td>,</td>
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<td><table class="frac" style="text-align:center;">
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<tr><td>1</td></tr>
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<tr><td class="overline">1110</td></tr>
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</table>
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</td>
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<td>,</td>
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</table>
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<table>
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<td><table class="frac" style="text-align:center;">
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<tr><td>1</td></tr>
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<tr><td class="overline">1111</td></tr>
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</table>
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</td>
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<td>,</td>
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<td><table class="frac" style="text-align:center;">
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<tr><td>1</td></tr>
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<tr><td class="overline">1112</td></tr>
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</table>
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</td>
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<td>,</td>
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<td><table class="frac" style="text-align:center;">
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<tr><td>1</td></tr>
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<tr><td class="overline">1113</td></tr>
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</table>
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</td>
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<td>,</td>
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<td><table class="frac" style="text-align:center;">
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<tr><td>1</td></tr>
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<tr><td class="overline">1114</td></tr>
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</table>
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</td>
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<td>,</td>
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<td><table class="frac" style="text-align:center;">
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<tr><td>1</td></tr>
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<tr><td class="overline">1115</td></tr>
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</table>
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</td>
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<td>,</td>
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<td><table class="frac" style="text-align:center;">
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<tr><td>1</td></tr>
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<tr><td class="overline">1116</td></tr>
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</table>
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</td>
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<td>,</td>
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<td><table class="frac" style="text-align:center;">
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<tr><td>1</td></tr>
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<tr><td class="overline">1117</td></tr>
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</table>
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</td>
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<td>,</td>
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<td><table class="frac" style="text-align:center;">
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<tr><td>1</td></tr>
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<tr><td class="overline">1118</td></tr>
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</table>
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</td>
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<td> and </td>
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<td><table class="frac" style="text-align:center;">
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<tr><td>1</td></tr>
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<tr><td class="overline">1119</td></tr>
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</table>
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</td>
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<td>.</td>
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</table>
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<P>
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This series converges as well.
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</P>
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<P>
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Find the value the series converges to.<BR />
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Give your answer rounded to 10 digits behind the decimal point.
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</p>
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