MaplePrimes - comments on Post, MRB Constant-C

MaplePrimes - comments on Post, MRB Constant-C http://www.mapleprimes.com/posts/37851-MRB-ConstantC en-us 2026 Maplesoft, A Division of Waterloo Maple Inc. Maplesoft Document System Wed, 10 Jun 2026 17:39:49 GMT Wed, 10 Jun 2026 17:39:49 GMT The latest comments added to the Post, MRB Constant-C http://www.mapleprimes.com/images/mapleprimeswhite.jpg MaplePrimes - comments on Post, MRB Constant-C http://www.mapleprimes.com/posts/37851-MRB-ConstantC Undefined http://www.mapleprimes.com/posts/37851-MRB-ConstantC?ref=Feed:MaplePrimes:MRB Constant-C:Comments#comment67398 The function f(x) = (-1)^x * x^(1/x) = exp(I*Pi*x + ln(x)/x) is not integrable on [1,infinity): note that f(x) ~ exp(I*Pi*x) as x -> infinity.  What does have a limit as N -> infinity (for integers N) is int(f(x), x = 1 .. 2*N) .  Similarly, sum(f(n),n=1..2*N) has a limit as N -> infinity.   The latest comments added to the Post, MRB Constant-C 67398 Sun, 22 Feb 2009 10:06:35 Z Robert Israel Robert Israel Point taken http://www.mapleprimes.com/posts/37851-MRB-ConstantC?ref=Feed:MaplePrimes:MRB Constant-C:Comments#comment67393 For x>0and f(x) = (-1)^x * x^(1/x) , the next document,<a href="../files/565_part%202%20sum%20vs%20int.mw"> </a><a href="/view.aspx?sf=67393/278026/part_2_sum_vs_int.mw">Download part_2_sum_vs_int.mw</a><a href="../files/565_part%202%20sum%20vs%20int.mw"></a> shows the difference of  and the ratio of abs(int(f(x), x = 1 .. 2*N))-1/2  and sum(f(n),n=1..2*N) as [even] integer N -> infinity. The difference seems to go to 0 and the ratio seems to go to 1. However, it would be nice to have a proof of some sort to show that they both converge upon the same number. I have given a simple geometric example of the summing of f in the following: <a href="http://marvinrayburns.com/what_is_mrb.mht">http://marvinrayburns.com/what_is_mrb.mht</a>. Perhaps a similar example of the integrating of f could help make the proof -- that is if it is true.   The MRB constant=0.187859... <form><input name="sequence" type="hidden" value="1"></form> <form> <table width="576" align="center"> <tbody> <tr> <td> <img style="vertical-align: -81;" src="/view.aspx?sf=67393/278026/426e3133f1e9ddc0cbb7b3511790ff17.gif" alt="" width="576" height="118" align="middle"> <table> <tbody> <tr valign="baseline"> <td> <img style="vertical-align: -6;" src="/view.aspx?sf=67393/278026/71a3b00eeea2bdc8b99ec364294684f5.gif" alt="" width="52" height="23"> </td> <td style="color: #000000; font-family: Times, serif; font-weight: bold; font-style: normal;" align="right">(1)</td> </tr> </tbody> </table> <img style="vertical-align: -6;" src="/view.aspx?sf=67393/278026/4dc69a1bc78f521ac6ec9141e0e6d7ab.gif" alt="" width="11" height="23"> <img style="vertical-align: -6;" src="/view.aspx?sf=67393/278026/2a7b08408f5b0eb49825a8db98b257ba.gif" alt="" width="11" height="23"> <img style="vertical-align: -6;" src="/view.aspx?sf=67393/278026/27a89efc8864878e10b681553ae9c090.gif" alt="" width="11" height="23"> <img style="vertical-align: -6;" src="/view.aspx?sf=67393/278026/e6b9338e1d2719339766757b486ae834.gif" alt="" width="11" height="23"> <img style="vertical-align: -6;" src="/view.aspx?sf=67393/278026/5e883fc4b6783980fedf5dc134db003c.gif" alt="" width="11" height="23"> <img style="vertical-align: -71;" src="/view.aspx?sf=67393/278026/b27beb11160bf4ee526010e46603bfa6.gif" alt="" width="576" height="131" align="middle"> <table> <tbody> <tr valign="baseline"> <td> <img style="vertical-align: -6;" src="/view.aspx?sf=67393/278026/4af598740e2c9a62e566d0ad89a4a163.gif" alt="" width="38" height="23"> </td> <td style="color: #000000; font-family: Times, serif; font-weight: bold; font-style: normal;" align="right">(2)</td> </tr> </tbody> </table> <img style="vertical-align: -6;" src="/view.aspx?sf=67393/278026/159622c32ebcf2cc747e6fa6a3304236.gif" alt="" width="11" height="23"> <img style="vertical-align: -6;" src="/view.aspx?sf=67393/278026/0db14390a4f6d3760b204f28740f6571.gif" alt="" width="11" height="23"> <img style="vertical-align: -6;" src="/view.aspx?sf=67393/278026/d5e9a1974f27e7f1e40ec1c3452f77e3.gif" alt="" width="11" height="23"> <img style="vertical-align: -6;" src="/view.aspx?sf=67393/278026/3ceb2cfa9b6c4990b8e0153b0f017b8a.gif" alt="" width="11" height="23"> <img style="vertical-align: -6;" src="/view.aspx?sf=67393/278026/a5a6bc44fa35277fbfa7b53e4f214a1b.gif" alt="" width="11" height="23"> <img style="vertical-align: -6;" src="/view.aspx?sf=67393/278026/6beb4ad9c34cbb8f46a52f04c8da6017.gif" alt="" width="11" height="23"> <img style="vertical-align: -6;" src="/view.aspx?sf=67393/278026/d71fa76042d82a76895efdb4197124aa.gif" alt="" width="11" height="23"> <img style="vertical-align: -6;" src="/view.aspx?sf=67393/278026/5e0a0aa25dad7dbb671f8a66a727c2c1.gif" alt="" width="11" height="23"> <img style="vertical-align: -6;" src="/view.aspx?sf=67393/278026/00accbce1bde87d4fc9e445ca8e08f70.gif" alt="" width="11" height="23"> <img style="vertical-align: -6;" src="/view.aspx?sf=67393/278026/1ab7ceb8395ce46383393971b1c4d463.gif" alt="" width="11" height="23"> <img style="vertical-align: -6;" src="/view.aspx?sf=67393/278026/6ec6b4053fa39dc479c5b83a44271eda.gif" alt="" width="11" height="23"> <img style="vertical-align: -6;" src="/view.aspx?sf=67393/278026/377147d6cf6153e47058bd54b72bc743.gif" alt="" width="11" height="23"> <img style="vertical-align: -6;" src="/view.aspx?sf=67393/278026/6f0526abee5a4f2ae200b85d4c27705f.gif" alt="" width="11" height="23"> <img style="vertical-align: -6;" src="/view.aspx?sf=67393/278026/492d62d8815109739f8e5e4be360c7ad.gif" alt="" width="11" height="23"> </td> </tr> </tbody> </table> <input name="sequence" type="hidden" value="1"></form> <a href="/view.aspx?sf=67393/278026/part_2_sum_vs_int.mw">Download part_2_sum_vs_int.mw</a>         <a href="/view.aspx?sf=67393/277092/074a0d60ee239843a19953a5a86f954b.gif"><img src="/view.aspx?sf=67393/277092/074a0d60ee239843a19953a5a86f954b.gif" alt=""></a><a href="/view.aspx?sf=67393/277092/17a31d0dabe993fe258efeac00d2186c.gif"><img src="/view.aspx?sf=67393/277092/17a31d0dabe993fe258efeac00d2186c.gif" alt=""></a><a href="/view.aspx?sf=67393/277092/257d966fc8651b42e9a7ee6e1f5fd676.gif"><img src="/view.aspx?sf=67393/277092/257d966fc8651b42e9a7ee6e1f5fd676.gif" alt=""></a><a href="/view.aspx?sf=67393/277092/27ece9d17ec32a7dea22cbaa4615e9b4.gif"><img src="/view.aspx?sf=67393/277092/27ece9d17ec32a7dea22cbaa4615e9b4.gif" alt=""></a><a href="/view.aspx?sf=67393/277092/46b3a7dea9677b8725781dea6d1e93e6.gif"><img src="/view.aspx?sf=67393/277092/46b3a7dea9677b8725781dea6d1e93e6.gif" alt=""></a><a href="/view.aspx?sf=67393/277092/602cd4ea6b88bcbc1b5285933f9b3d1b.gif"><img src="/view.aspx?sf=67393/277092/602cd4ea6b88bcbc1b5285933f9b3d1b.gif" alt=""></a><a href="/view.aspx?sf=67393/277092/a055c8a927e424077648b004609bd8b1.gif"><img src="/view.aspx?sf=67393/277092/a055c8a927e424077648b004609bd8b1.gif" alt=""></a> The latest comments added to the Post, MRB Constant-C 67393 Tue, 24 Feb 2009 01:05:29 Z Marvin Ray Burns Marvin Ray Burns not true http://www.mapleprimes.com/posts/37851-MRB-ConstantC?ref=Feed:MaplePrimes:MRB Constant-C:Comments#comment82069 My conjecture is false. See<a href="http://marvinrayburns.com/latest.html"> http://marvinrayburns.com/latest.html. </a>   The latest comments added to the Post, MRB Constant-C 82069 Sun, 08 Mar 2009 03:30:10 Z Marvin Ray Burns Marvin Ray Burns Some research done http://www.mapleprimes.com/posts/37851-MRB-ConstantC?ref=Feed:MaplePrimes:MRB Constant-C:Comments#comment81795 It appears that RICHARD J. MATHAR has written a paper on this subject  of Int(f(x)) where f(x) = (-1)^x * x^(1/x) = exp(I*Pi*x + ln(x)/x).  <a href="http://arxiv.org/PS_cache/arxiv/pdf/0912/0912.3844v2.pdf">http://arxiv.org/PS_cache/arxiv/pdf/0912/0912.3844v2.pdf</a>. He develops three new acceleration methods for computing the integral. The latest comments added to the Post, MRB Constant-C 81795 Sat, 24 Apr 2010 23:03:38 Z Marvin Ray Burns Marvin Ray Burns The MKB constant http://www.mapleprimes.com/posts/37851-MRB-ConstantC?ref=Feed:MaplePrimes:MRB Constant-C:Comments#comment88336 I call that value, the integral over (-1)^x*x^(1/x) between 1 and infinity, the MKB constant.    It is Sloane's<a href="http://www.research.att.com/~njas/sequences/A157852"> </a><a href="https://oeis.org/A157852">A157852.</a> The latest comments added to the Post, MRB Constant-C 88336 Sun, 09 May 2010 06:25:41 Z Marvin Ray Burns Marvin Ray Burns so what is the question or task? http://www.mapleprimes.com/posts/37851-MRB-ConstantC?ref=Feed:MaplePrimes:MRB Constant-C:Comments#comment88338 <pre> Is it to (numercial) compute Lim( Int( exp( I*Pi*x + ln(x)/x ), x= 1 .. 2*n ), n = infinity ) according to Robert Israel's comment or what? </pre> The latest comments added to the Post, MRB Constant-C 88338 Sun, 09 May 2010 15:02:17 Z Axel Vogt Axel Vogt