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 fs=1/21/2*x, or implicitly, f=s+1/2(1x)
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Since f=s+1/2(1x) it is also true that1/2*x1/2=sf, so what can we say about the following graph?
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Download april092011.mw
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(1x) one might expect Max(f)=Max(s)+1/2(1x), 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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