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Cell["\<\

Trying to verify or otherwise generate a value for these sums.\
\>", "Text"],

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Cell["\<\

Let me to try this one step at a time.  Verify (W-7).\
\>", "Text"],

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Alternating series--some end up below the result below, some above.  So far, \
it looks good!\
\>", "Text"],

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Very good.\
\>", "Text"],

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Cell["\<\
Now what if I considered just one of my two 1/x factors form (W-8).  It looks \
like this converges really quickly!
    \
\>", "Text"],

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Cell["\<\

Now, the issue with creating this through integration is that I need to kill \
off the odd Taylor components before integrating, not after.  Note that I \
need the power on each particular x to be equal to (2m - 2n + 1) - 1.  In \
other words, I need to adjust each one by -2n.  To be fair, however, the \
primary issue at hand is only the Taylor term that will correspond to 1/x \
before integration.  If I subtracted out that term then integrated, divided \
and evaluated, it would give the same result.  Let me give it a shot.

Start then with my series V(1/p,x).

Multiply it by x^(-2n)

Subtract x^-1 times the ( 2n - 1) th Taylor coefficient.

Integrate

Divide it by x^(-2n+1) to adjust it again

Then take the real part evaluated at x=i.\
\>", "Text"],

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So this looks really, really good.\
\>", "Text"],

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Cell["\<\

This MUST give me the difference, when evaluated at x=0.\
\>", "Text"],

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Cell["\<\

Fine. It doesn't want to simplify this to a state where I can evaluate it at \
x=0.  I can, however, examine the series above and see that for this \
particular choice of series, I always end up with an extra -1/7.\
\>", "Text"],

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Cell["\<\

Very good.  I see then that this works for the addition of one of the series. \
 Now I should be able to repeat the procedure above for the other series.  \
First, though, let me see if this is simplifiable.\
\>", "Text"],

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Fine.  It doesn't want to simplify this.  At least I know it works--maybe Dr. \
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I'm pretty sure the above integral is right.  I just need to figure out how \
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out.  In this particular expansion, it will have subtracted from it the value \
of the coefficient on the m=0 item!  
\
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