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## A Very False Graph

Maple

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.

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

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

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However, their graphs have exactly the same shape.

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

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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?

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With 20 digits of precision there is no aliasing.

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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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