I wanted to see what LErchPhi function has regarding this sum for a good reason,
The sum itself (without the 10^n) is sum(1/(4*n^2-4*n+4*100000000^2+1),n=1..infinity).
Looks ordinary but it is not , it is Pi , not exactly Pi for real but for all practical purpose it is
because the real answer is A * B * tanh(Pi*c) , here is an example but lower precision
sum(1/(4*n^2-4*n+4*100^2+1),n=1..infinity) = 1/800*Pi*tanh(100*Pi), if we correct the 800 term
we have then Pi*tanh(100*Pi) and this is where it is funny because this number is very close
to Pi up to the 272'th digits. So if we increase the 1000...0000 factor we can get ANY precision.
Normaly that kind of infinite sum has to do with the Psi function BUT when the arguments
are complex the 2 complex values collide and gives something completely unexpeected.
You see the point ? Unfortunately by adding 10^n in the process mixes this singularity.
Too bad. In general the LerchPhi function is <unknown> for specific arguments, there is
a lengthly litterature on it BUT very few examples of explicit values. LerchPhi(z, a, 1) is rather
simple when the argument is 1 but not when it is something else.