A Very False Graph

April 09 2011 Marvin Ray Burns 485
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 It seems I can't add a response to this message, so I added some detail to it.

Consider f, the partial sums of the convergent series related toas n goes to infinity for continuous z>0.

(1)

Also consider s, the partial sums of the divergent series related to as n goes to infinity for continuous z>0.

(2)

Obviously the graphs of f and s intersect at only one point.

(3)

However, their graphs have exactly the same shape.

 

 

 

It is natural to inquire as to the difference between f and s.

(4)

The table above is very reveling. It indicates that f-s=1/2-1/2*x, or implicitly, f=s+1/2(1-x)

 
 

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Since f=s+1/2(1-x) it is also true that1/2*x-1/2=s-f, so what can we say about the following graph?

 



Download april092011.mw

 

With 20 digits of precision there is no aliasing.

(1)

 

With as few as 14 or 15 digits of precision there is no visible aliasing.

 

 

 

Since  f=s+1/2(1-x) one might expect  Max(f)=Max(s)+1/2(1-x), but such is not the case. Consider
the extrema of f and s:

We see f has a max of approx 199.6486735 at 22.8766445, and
s has a max of approx 211.4872368 at 22.9359616.

The max of s is approx 11.8385633 greater that that of f.

Notice that f and s have their max at different values of the independent variable;
s has its max approx  0.5931714 to the left of that for f.

 

 

After taking the second derivative, numeric evaluation indicates
that f and s both have an inflection point at z=20.0931628620845....

 

marvinrayburns.com


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