MaplePrimes

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    <title>MaplePrimes - comments on Post, The MRB Constant</title>
    <link>http://www.mapleprimes.com/posts/42157-The-MRB-Constant</link>
    <language>en-us</language>
    <copyright>2026 Maplesoft, A Division of Waterloo Maple Inc.</copyright>
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    <lastBuildDate>Fri, 12 Jun 2026 04:25:51 GMT</lastBuildDate>
    <pubDate>Fri, 12 Jun 2026 04:25:51 GMT</pubDate>
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    <itunes:summary />
    <description>The latest comments added to the Post, The MRB Constant</description>
    <image>
      <url>http://www.mapleprimes.com/images/mapleprimeswhite.jpg</url>
      <title>MaplePrimes - comments on Post, The MRB Constant</title>
      <link>http://www.mapleprimes.com/posts/42157-The-MRB-Constant</link>
    </image>
    <item>
      <title>Updated</title>
      <link>http://www.mapleprimes.com/posts/42157-The-MRB-Constant?ref=Feed:MaplePrimes:The MRB Constant:Comments#comment78786</link>
      <itunes:summary>I updated your post with the contents of your worksheet. I have also placed it on the front page of MaplePrimes.

&lt;p&gt;____&lt;br/&gt;
William Spaetzel&lt;br/&gt;
Marketing Engineer, Maplesoft&lt;/p&gt;</itunes:summary>
      <description>The latest comments added to the Post, The MRB Constant</description>
      <guid>78786</guid>
      <pubDate>Tue, 12 Dec 2006 00:53:29 Z</pubDate>
      <itunes:author>Will</itunes:author>
      <author>Will</author>
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    <item>
      <title>curious</title>
      <link>http://www.mapleprimes.com/posts/42157-The-MRB-Constant?ref=Feed:MaplePrimes:The MRB Constant:Comments#comment78780</link>
      <itunes:summary>I must have missed something in this post. Perhaps you could help me understand what some of the insights are?

You started off with something like,

F1:=sum((-1)^n*(n^(1/n)-1),n=1..x):
F2:=sum((-1)^n*(n^(1/n)-2),n=1..x):
F3:=sum((-1)^n*(n^(1/n)-3),n=1..x):

And then you noticed that for each second set of values for these, ie.for x=2,4,6, the F1, F2, and F3 would agree.

But that's just because when x is any even positive integer all three of F1, F2, and F3 simplify immediately to  sum((-1)^n*n^(1/n),n = 1 .. infinity);

Now, Maple can show that directly, with,
simplify([F1,F2,F3]) assuming x::even;

You wrote, "Notice that the blue, red and green graphs meet at least one y-value. As x goes to infinity, that y-value is the value of the MRB Constant." But isn't that just the same as the limits of any subsequences of the function coloured green itself, as x goes to infinity? What do the red and blue curves add, for illustrating the behaviour of the green curve?

So it seems that you've named  sum((-1)^n*n^(1/n),n=1..infinity) . Is that right? 

What do you think of the value that Maple gives when one takes evalf() of that expression? Do you think that the green curve has some subsequences that converge to both a positive number and a negative number? If so, how do you think that those relate to the evalf() result?

I couldn't follow what the summations mean when the index ranges from 1 to sqrt(3). I had thought that the dummy index n was representing positive integers. What does this mean,
sum((-1)^n*n^(1/n),n = 1 .. 27*6^(1/2))));

Thanks very much,
acer
</itunes:summary>
      <description>The latest comments added to the Post, The MRB Constant</description>
      <guid>78780</guid>
      <pubDate>Tue, 12 Dec 2006 12:45:37 Z</pubDate>
      <itunes:author>acer</itunes:author>
      <author>acer</author>
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    <item>
      <title>actually</title>
      <link>http://www.mapleprimes.com/posts/42157-The-MRB-Constant?ref=Feed:MaplePrimes:The MRB Constant:Comments#comment78778</link>
      <itunes:summary>So, since the limit(n^(1/n),n=infinity) is 1, do you suspect that partial sums of the alternating series (-1)^n*n^(1/n) would necessarily have two limiting values (in terms of the alternating partial sums)?

How do you suspect that this might relate, if at all, to the value returned from,
evalf(sum((-1)^n*n^(1/n),n=1..infinity));

Maybe there already is a body of theory for this sort of thing.

What do you foresee a theory about this looking like? I ask because, once one has been motivated by some casual observations, the next step in mathematics is to develop a formalism and a notation. It's not easy to get a good notation, but a sure sign of success is that good notation leads to rich insight. I'm not sure that insight can come from observation alone.

acer
</itunes:summary>
      <description>The latest comments added to the Post, The MRB Constant</description>
      <guid>78778</guid>
      <pubDate>Tue, 12 Dec 2006 23:21:52 Z</pubDate>
      <itunes:author>acer</itunes:author>
      <author>acer</author>
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    <item>
      <title>acer, you wrote:&gt;I couldn't</title>
      <link>http://www.mapleprimes.com/posts/42157-The-MRB-Constant?ref=Feed:MaplePrimes:The MRB Constant:Comments#comment86095</link>
      <itunes:summary>acer, you wrote:
&gt;I couldn't follow what the summations mean when the index ranges from 1 to sqrt(3). I had 
&gt;thought that the dummy index n was representing positive integers. What does this mean,
&gt;sum((-1)^n*n^(1/n),n = 1 .. 27*6^(1/2))));
It is important to know, there I am just exploring to see what value
is represented by the graphs that I intend to research next.


You wrote:
&gt;But isn't that just the same as the limits of any subsequences of the function coloured 
&gt;green itself, as x goes to infinity? What do the red and blue curves add, for illustrating 
&gt;the behaviour of the green curve?
Yes you are right in that it is the top of the green graph. However, as
I demonstrated in the above table, the top value graphed in green is shared
(every other term) by the red and blue. 
That sharig of values is because, as you correcly wrote,
&gt;But that's just because when x is any even positive integer all three of F1, F2, and F3
&gt;simplify immediately to sum((-1)^n*n^(1/n),n = 1 .. infinity);

The point of the post is not that it is hard to understand.
The point is, my books failed to tell me that there exists rich value
in breaking up some non-converging infinite series into two series that
do converge. As far as my point is concerned, I only have
scratched the surface. I see (and foresee) that in all disciplines, where there
exists a model that has non-converging alternating sums, much progress can be made
in breaking up the model into series that do indeed converge.



I hope this helps,
From the study bench of
Marvin Ray Burns
(317) 371-6571
P.O. Box 19785
Indianapolis, IN 46219
</itunes:summary>
      <description>The latest comments added to the Post, The MRB Constant</description>
      <guid>86095</guid>
      <pubDate>Tue, 12 Dec 2006 23:49:42 Z</pubDate>
      <itunes:author>Marvin Ray Burns</itunes:author>
      <author>Marvin Ray Burns</author>
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      <title>One Member of the Family is an Absolutely Converging Series</title>
      <link>http://www.mapleprimes.com/posts/42157-The-MRB-Constant?ref=Feed:MaplePrimes:The MRB Constant:Comments#comment78777</link>
      <itunes:summary>&lt;p&gt;I show in the worksheet, mrbgraphs a, that the constant of concern (the MRB Constant) is indeed a limiting point at Infinity for one member of the family. Since all members have equal partial sums at even intervals, they all converge to the mrb in the following sense.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt; &lt;br&gt; &lt;br&gt;&lt;/strong&gt;&lt;/p&gt;
&lt;form name="worksheet_form"&gt;
&lt;table style="width: 576px;" align="center"&gt;
&lt;tbody&gt;
&lt;tr&gt;
&lt;td&gt;
&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"&gt;&lt;span style="color: #000000; font-size: 100%; font-family: Times New Roman,serif; font-weight: bold; font-style: normal;"&gt;Last update, Firday Dec 17, 7:17PM&lt;/span&gt;&lt;/p&gt;
&lt;table style="margin-left: 0px; margin-right: 0px;"&gt;
&lt;tbody&gt;
&lt;tr valign="baseline"&gt;
&lt;td&gt;&lt;span style="color: #ff0000; font-size: 100%; font-family: monospace,monospace; font-weight: bold; font-style: normal;"&gt; &amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
&lt;td&gt;
&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"&gt;&lt;span style="color: #ff0000; font-size: 100%; font-family: monospace,monospace; font-weight: bold; font-style: normal;"&gt;restart;&lt;/span&gt;&lt;/p&gt;
&lt;/td&gt;
&lt;/tr&gt;
&lt;/tbody&gt;
&lt;/table&gt;
&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"&gt;&lt;span style="color: #000000; font-size: 100%; font-family: Times New Roman,serif; font-weight: normal; font-style: normal;"&gt;&amp;nbsp; &lt;/span&gt;&lt;/p&gt;
&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="center"&gt;&lt;span style="color: #000000; font-size: 100%; font-family: Times New Roman,serif; font-weight: normal; font-style: normal;"&gt;&amp;nbsp; &lt;/span&gt;&lt;/p&gt;
&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="center"&gt;&lt;span style="color: #000000; font-size: 100%; font-family: Times New Roman,serif; font-weight: normal; font-style: normal;"&gt;&amp;nbsp; &lt;/span&gt;&lt;/p&gt;
&lt;table style="margin-left: 0px; margin-right: 0px;"&gt;
&lt;tbody&gt;
&lt;tr valign="baseline"&gt;
&lt;td&gt;&lt;span style="color: #ff0000; font-size: 100%; font-family: monospace,monospace; font-weight: bold; font-style: normal;"&gt; &amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
&lt;td&gt;
&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"&gt;&lt;span style="color: #ff0000; font-size: 100%; font-family: monospace,monospace; font-weight: bold; font-style: normal;"&gt;f1:=x-&amp;gt;sum((-1)^n*(n^(1/n)-1),n=1..x);&lt;/span&gt;&lt;/p&gt;
&lt;/td&gt;
&lt;/tr&gt;
&lt;/tbody&gt;
&lt;/table&gt;
&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="center"&gt;&lt;img style="vertical-align: -20;" src="/view.aspx?sf=78777/427927/569b31e060d274e9e9695b291b0b6933.gif" alt="f1 := proc (x) options operator, arrow; sum((-1)^n*(n^(1/n)-1), n = 1 .. x) end proc" width="191" height="57"&gt;&lt;/p&gt;
&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"&gt;&lt;span style="color: #000000; font-size: 100%; font-family: Times New Roman,serif; font-weight: normal; font-style: normal;"&gt;Technically, as x goes to infinity, f1(x) does converge. n^(1/n)-&amp;gt;0 as n-&amp;gt;infinity. Therefore, the function that describes the local maximums of f:&lt;/span&gt;&lt;/p&gt;
&lt;table style="margin-left: 0px; margin-right: 0px;"&gt;
&lt;tbody&gt;
&lt;tr valign="baseline"&gt;
&lt;td&gt;&lt;span style="color: #ff0000; font-size: 100%; font-family: Times New Roman,serif; font-weight: normal; font-style: normal;"&gt; &amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
&lt;td&gt;
&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"&gt;&lt;img style="vertical-align: -20;" src="/view.aspx?sf=78777/427927/8d7f36ad5b353ced21f94f0986e1c22c.gif" alt="limit(sum((-1)^n*n^(1/n), n = 1 .. 2*Inf), Inf = infinity)" width="175" height="57"&gt;&lt;/p&gt;
&lt;/td&gt;
&lt;/tr&gt;
&lt;/tbody&gt;
&lt;/table&gt;
&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"&gt;&lt;span style="color: #000000; font-size: 100%; font-family: Times New Roman,serif; font-weight: normal; font-style: normal;"&gt;aslo converges.&lt;/span&gt;&lt;/p&gt;
&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"&gt;&lt;span style="color: #000000; font-size: 100%; font-family: Times New Roman,serif; font-weight: normal; font-style: normal;"&gt;Because the function that describes the local maximums of f converges and because the limit(x^1/x,x=infinity)=1, &amp;nbsp;the function that describes the local minimums of f:&lt;/span&gt;&lt;/p&gt;
&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"&gt;&lt;span style="color: #000000; font-size: 100%; font-family: Times New Roman,serif; font-weight: normal; font-style: normal;"&gt;&amp;nbsp; &lt;/span&gt;&lt;/p&gt;
&lt;table style="margin-left: 0px; margin-right: 0px;"&gt;
&lt;tbody&gt;
&lt;tr valign="baseline"&gt;
&lt;td&gt;&lt;span style="color: #ff0000; font-size: 100%; font-family: Times New Roman,serif; font-weight: normal; font-style: normal;"&gt; &amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
&lt;td&gt;
&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"&gt;&lt;img style="vertical-align: -20;" src="/view.aspx?sf=78777/427927/68099efb5ecd287ddcbbe26fde77ce8d.gif" alt="limit(sum((-1)^n*n^(1/n), n = 1 .. 2*Inf+1), Inf = infinity)" width="194" height="57"&gt;&lt;/p&gt;
&lt;/td&gt;
&lt;/tr&gt;
&lt;/tbody&gt;
&lt;/table&gt;
&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"&gt;&lt;span style="color: #000000; font-size: 100%; font-family: Times New Roman,serif; font-weight: normal; font-style: normal;"&gt;also converges.&lt;/span&gt;&lt;/p&gt;
&lt;table style="margin-left: 0px; margin-right: 0px;"&gt;
&lt;tbody&gt;
&lt;tr valign="baseline"&gt;
&lt;td&gt;&lt;span style="color: #ff0000; font-size: 100%; font-family: monospace,monospace; font-weight: bold; font-style: normal;"&gt; &amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
&lt;td&gt;
&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"&gt;&lt;span style="color: #ff0000; font-size: 100%; font-family: monospace,monospace; font-weight: bold; font-style: normal;"&gt;&amp;nbsp; &lt;/span&gt;&lt;/p&gt;
&lt;/td&gt;
&lt;/tr&gt;
&lt;/tbody&gt;
&lt;/table&gt;
&lt;table style="margin-left: 0px; margin-right: 0px;"&gt;
&lt;tbody&gt;
&lt;tr valign="baseline"&gt;
&lt;td&gt;&lt;span style="color: #ff0000; font-size: 100%; font-family: monospace,monospace; font-weight: bold; font-style: normal;"&gt; &amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
&lt;td&gt;
&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"&gt;&lt;span style="color: #ff0000; font-size: 100%; font-family: monospace,monospace; font-weight: bold; font-style: normal;"&gt;&amp;nbsp;limit(x^1/x,x=ininity);&lt;/span&gt;&lt;/p&gt;
&lt;/td&gt;
&lt;/tr&gt;
&lt;/tbody&gt;
&lt;/table&gt;
&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="center"&gt;&lt;img style="vertical-align: -6;" src="/view.aspx?sf=78777/427927/2f7b9a3e36b87f4f1c3386b99f23ddb3.gif" alt="1" width="13" height="23"&gt;&lt;/p&gt;
&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"&gt;&lt;span style="color: #000000; font-size: 100%; font-family: Times New Roman,serif; font-weight: normal; font-style: normal;"&gt;As for the plots I was concerned about in mrbgraphs 1, I can focus solely upon f1 since we know it converges. I will look at the real and imaginary parts and the absolute value of f1.&lt;/span&gt;&lt;/p&gt;
&lt;table style="margin-left: 0px; margin-right: 0px;"&gt;
&lt;tbody&gt;
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&lt;td&gt;&lt;span style="color: #ff0000; font-size: 100%; font-family: monospace,monospace; font-weight: bold; font-style: normal;"&gt; &amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
&lt;td&gt;
&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"&gt;&lt;span style="color: #ff0000; font-size: 100%; font-family: monospace,monospace; font-weight: bold; font-style: normal;"&gt;with(plots):&lt;/span&gt;&lt;/p&gt;
&lt;/td&gt;
&lt;/tr&gt;
&lt;/tbody&gt;
&lt;/table&gt;
&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"&gt;&lt;span style="color: #0000ff; font-size: 83%; font-family: monospace,monospace; font-weight: normal; font-style: normal;"&gt;Warning, the name changecoords has been redefined&lt;br&gt; &lt;/span&gt;&lt;/p&gt;
&lt;table style="margin-left: 0px; margin-right: 0px;"&gt;
&lt;tbody&gt;
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&lt;td&gt;&lt;span style="color: #ff0000; font-size: 100%; font-family: monospace,monospace; font-weight: bold; font-style: normal;"&gt; &amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
&lt;td&gt;
&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"&gt;&lt;span style="color: #ff0000; font-size: 100%; font-family: monospace,monospace; font-weight: bold; font-style: normal;"&gt;Inf:=20:&lt;br&gt; &lt;/span&gt;&lt;span style="color: #ff0000; font-size: 100%; font-family: monospace,monospace; font-weight: bold; font-style: normal;"&gt;&amp;nbsp;P0(Inf):=plot(Re(f1(x)),x=1..Inf,color=green,tickmarks=[21,3],style=point):&lt;br&gt; &lt;/span&gt;&lt;span style="color: #ff0000; font-size: 100%; font-family: monospace,monospace; font-weight: bold; font-style: normal;"&gt;&amp;nbsp;P1(Inf):=plot(Im(f1(x)),x=1..Inf,color=blue,tickmarks=[21,3],style=point):&lt;br&gt; &lt;/span&gt;&lt;span style="color: #ff0000; font-size: 100%; font-family: monospace,monospace; font-weight: bold; font-style: normal;"&gt;&amp;nbsp;P2(Inf):=plot(abs(f1(x)),x=1..Inf,color=red,tickmarks=[21,3],thickness=3):&lt;br&gt; &lt;/span&gt;&lt;span style="color: #ff0000; font-size: 100%; font-family: monospace,monospace; font-weight: bold; font-style: normal;"&gt;&amp;nbsp;display([P0(Inf),P1(Inf),P2(Inf)]);&lt;br&gt; &lt;/span&gt;&lt;span style="color: #ff0000; font-size: 100%; font-family: monospace,monospace; font-weight: bold; font-style: normal;"&gt;&lt;br&gt; &lt;/span&gt;&lt;/p&gt;
&lt;/td&gt;
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&lt;table style="margin-left: 0px; margin-right: 0px;"&gt;
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&lt;td&gt;&lt;span style="color: #ff0000; font-size: 100%; font-family: monospace,monospace; font-weight: bold; font-style: normal;"&gt; &amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
&lt;td&gt;
&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"&gt;&lt;span style="color: #ff0000; font-size: 100%; font-family: monospace,monospace; font-weight: bold; font-style: normal;"&gt;&amp;nbsp; &lt;/span&gt;&lt;/p&gt;
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&lt;td&gt;&lt;span style="color: #ff0000; font-size: 100%; font-family: monospace,monospace; font-weight: bold; font-style: normal;"&gt; &amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
&lt;td&gt;
&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"&gt;&lt;span style="color: #ff0000; font-size: 100%; font-family: monospace,monospace; font-weight: bold; font-style: normal;"&gt;&amp;nbsp; &lt;/span&gt;&lt;/p&gt;
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&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="center"&gt;&lt;a href="http://www.maplesoft.com/support/faqs/MapleNet/redirect.aspx?param=plot_java_14206"&gt;&lt;img style="border: none;" src="/view.aspx?sf=78777/427927/4f3ce3d2487ceaf478a2699409a92368.gif" alt="" width="576" height="576" align="middle"&gt;&lt;/a&gt;&lt;/p&gt;
&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"&gt;&lt;span style="color: #000000; font-size: 100%; font-family: Times New Roman,serif; font-weight: normal; font-style: normal;"&gt;From the red graph, it appears that abs(f1) has local minimums (mins) at every odd value of x and the value of a min is getting closer to the value of the next odd value as the value of x grows without bound .&lt;/span&gt;&lt;/p&gt;
&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"&gt;&lt;span style="color: #000000; font-size: 100%; font-family: Times New Roman,serif; font-weight: normal; font-style: normal;"&gt;By definition, the red graph, the absolute value of f1 is simply sqrt((-1/2*sin(n*Pi))^2+Re(f1(n))^2); in maple's natural simplification: (I say nothing esle about that.)&lt;/span&gt;&lt;/p&gt;
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&lt;td&gt;&lt;span style="color: #ff0000; font-size: 100%; font-family: monospace,monospace; font-weight: bold; font-style: normal;"&gt; &amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
&lt;td&gt;
&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"&gt;&lt;span style="color: #ff0000; font-size: 100%; font-family: monospace,monospace; font-weight: bold; font-style: normal;"&gt;ff1:=sqrt((-1/2*sin(n*Pi))^2+Re(f1(n))^2);&lt;/span&gt;&lt;/p&gt;
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&lt;td&gt;&lt;span style="color: #ff0000; font-size: 100%; font-family: monospace,monospace; font-weight: bold; font-style: normal;"&gt; &amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
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&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"&gt;&lt;span style="color: #ff0000; font-size: 100%; font-family: monospace,monospace; font-weight: bold; font-style: normal;"&gt;&amp;nbsp; &lt;/span&gt;&lt;/p&gt;
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&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="center"&gt;&lt;img style="vertical-align: -20;" src="/view.aspx?sf=78777/427927/751cee5c948c84a42fd2365c5e602a55.gif" alt="ff1 := (1/2)*(sin(n*Pi)^2+4*(1/2+(1/2)*Re((-1)^(n+1)+2*(sum((-1)^n*n^(1/n), n = 1 .. n))))^2)^(1/2)" width="478" height="65"&gt;&lt;/p&gt;
&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"&gt;&lt;span style="color: #000000; font-size: 100%; font-family: Times New Roman,serif; font-weight: normal; font-style: normal;"&gt;Looking further along the x-axis of &amp;nbsp;graph we see that those mins head twoard the MRB Constant, here called f .&lt;/span&gt;&lt;/p&gt;
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&lt;td&gt;&lt;span style="color: #ff0000; font-size: 100%; font-family: monospace,monospace; font-weight: bold; font-style: normal;"&gt; &amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
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&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"&gt;&lt;span style="color: #ff0000; font-size: 100%; font-family: monospace,monospace; font-weight: bold; font-style: normal;"&gt;&amp;nbsp; &lt;/span&gt;&lt;/p&gt;
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&lt;td&gt;&lt;span style="color: #ff0000; font-size: 100%; font-family: monospace,monospace; font-weight: bold; font-style: normal;"&gt; &amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
&lt;td&gt;
&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"&gt;&lt;span style="color: #ff0000; font-size: 100%; font-family: monospace,monospace; font-weight: bold; font-style: normal;"&gt;f:=sum((-1)^n*(n^(1/n)-1),n=1..infinity):&lt;br&gt; &lt;/span&gt;&lt;span style="color: #ff0000; font-size: 100%; font-family: monospace,monospace; font-weight: bold; font-style: normal;"&gt;Inf:=200:&lt;br&gt; &lt;/span&gt;&lt;span style="color: #ff0000; font-size: 100%; font-family: monospace,monospace; font-weight: bold; font-style: normal;"&gt;&lt;br&gt; &lt;/span&gt;&lt;span style="color: #ff0000; font-size: 100%; font-family: monospace,monospace; font-weight: bold; font-style: normal;"&gt;&amp;nbsp;&amp;nbsp;mrb:=plot(f,x=120..Inf,color=black):&lt;br&gt; &lt;/span&gt;&lt;span style="color: #ff0000; font-size: 100%; font-family: monospace,monospace; font-weight: bold; font-style: normal;"&gt;&amp;nbsp;&amp;nbsp;P0(Inf):=plot(Re(f1(x)),x=120..Inf,color=green):&lt;br&gt; &lt;/span&gt;&lt;span style="color: #ff0000; font-size: 100%; font-family: monospace,monospace; font-weight: bold; font-style: normal;"&gt;&amp;nbsp;&amp;nbsp;P1(Inf):=plot(Im(f1(x)),x=120..Inf,color=blue):&lt;br&gt; &lt;/span&gt;&lt;span style="color: #ff0000; font-size: 100%; font-family: monospace,monospace; font-weight: bold; font-style: normal;"&gt;&amp;nbsp;&amp;nbsp;P2(Inf):=plot(abs(f1(x)),x=120..Inf,color=red,thickness=2):&lt;br&gt; &lt;/span&gt;&lt;span style="color: #ff0000; font-size: 100%; font-family: monospace,monospace; font-weight: bold; font-style: normal;"&gt;&lt;br&gt; &lt;/span&gt;&lt;span style="color: #ff0000; font-size: 100%; font-family: monospace,monospace; font-weight: bold; font-style: normal;"&gt;#3 graphs zooming in on the MRB:&lt;br&gt; &lt;/span&gt;&lt;span style="color: #ff0000; font-size: 100%; font-family: monospace,monospace; font-weight: bold; font-style: normal;"&gt;&amp;nbsp;&amp;nbsp;&amp;nbsp;display([mrb,P0(Inf),P1(Inf),P2(Inf)]);&lt;br&gt; &lt;/span&gt;&lt;span style="color: #ff0000; font-size: 100%; font-family: monospace,monospace; font-weight: bold; font-style: normal;"&gt;&amp;nbsp;&amp;nbsp;&amp;nbsp;display([mrb,P0(Inf),P1(Inf),P2(Inf)],view=[120..200,evalf(f-1/2^2)..evalf(f+1/2^2)]);&lt;br&gt; &lt;/span&gt;&lt;span style="color: #ff0000; font-size: 100%; font-family: monospace,monospace; font-weight: bold; font-style: normal;"&gt;&amp;nbsp;&amp;nbsp;&amp;nbsp;display([mrb,P0(Inf),P1(Inf),P2(Inf)],view=[180..200,evalf(f-1/2^3)..evalf(f+1/2^3)]);&lt;br&gt; &lt;/span&gt;&lt;/p&gt;
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&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="center"&gt;&lt;a href="http://www.maplesoft.com/support/faqs/MapleNet/redirect.aspx?param=plot_java_14206"&gt;&lt;img style="border: none;" src="/view.aspx?sf=78777/427927/bdbaa6aa92101238cda54254f4d0feea.gif" alt="" width="576" height="305" align="middle"&gt;&lt;/a&gt;&lt;/p&gt;
&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="center"&gt;&lt;a href="http://www.maplesoft.com/support/faqs/MapleNet/redirect.aspx?param=plot_java_14206"&gt;&lt;img style="border: none;" src="/view.aspx?sf=78777/427927/07d7ddccd74dd05885f4341d26a0daf2.gif" alt="" width="400" height="400" align="middle"&gt;&lt;/a&gt;&lt;/p&gt;
&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="center"&gt;&lt;a href="http://www.maplesoft.com/support/faqs/MapleNet/redirect.aspx?param=plot_java_14206"&gt;&lt;img style="border: none;" src="/view.aspx?sf=78777/427927/513c8964f55fe4106f4df84286bbee75.gif" alt="" width="400" height="400" align="middle"&gt;&lt;/a&gt;&lt;/p&gt;
&lt;table style="margin-left: 0px; margin-right: 0px;"&gt;
&lt;tbody&gt;
&lt;tr valign="baseline"&gt;
&lt;td&gt;&lt;span style="color: #ff0000; font-size: 100%; font-family: monospace,monospace; font-weight: bold; font-style: normal;"&gt; &amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
&lt;td&gt;
&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"&gt;&lt;span style="color: #ff0000; font-size: 100%; font-family: monospace,monospace; font-weight: bold; font-style: normal;"&gt;&amp;nbsp; &lt;/span&gt;&lt;/p&gt;
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&lt;/table&gt;
&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"&gt;&lt;span style="color: #000000; font-size: 100%; font-family: Times New Roman,serif; font-weight: normal; font-style: normal;"&gt;&amp;nbsp; &lt;/span&gt;&lt;/p&gt;
&lt;/td&gt;
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&lt;/table&gt;
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&lt;p&gt;&lt;strong&gt;&lt;br&gt; &lt;br&gt;&lt;/strong&gt;&lt;/p&gt;
&lt;p&gt;&lt;a href="/view.aspx?sf=78777/427927/565_mrbgraphs_a.mw"&gt;Download 565_mrbgraphs_a.mw&lt;/a&gt;&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;&lt;br&gt;&lt;/strong&gt;&lt;/p&gt;</itunes:summary>
      <description>The latest comments added to the Post, The MRB Constant</description>
      <guid>78777</guid>
      <pubDate>Wed, 13 Dec 2006 03:18:19 Z</pubDate>
      <itunes:author>Marvin Ray Burns</itunes:author>
      <author>Marvin Ray Burns</author>
    </item>
    <item>
      <title>Updated and MaplePrimes bug</title>
      <link>http://www.mapleprimes.com/posts/42157-The-MRB-Constant?ref=Feed:MaplePrimes:The MRB Constant:Comments#comment86093</link>
      <itunes:summary>I replaced your images with an HTML version of your worksheet.

If you upload your worksheets in .mw format, MaplePrimes will give you the HTML code that you can paste in. Much easier than uploading screenshots.

You also seem to have found a bug with the site, when you upload in image with a space in the filename, the resizing script does not work correctly. I will fix that. 

&lt;p&gt;____&lt;br/&gt;
William Spaetzel&lt;br/&gt;
Marketing Engineer, Maplesoft&lt;/p&gt;</itunes:summary>
      <description>The latest comments added to the Post, The MRB Constant</description>
      <guid>86093</guid>
      <pubDate>Wed, 13 Dec 2006 03:23:37 Z</pubDate>
      <itunes:author>Will</itunes:author>
      <author>Will</author>
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