[seqfan] Curious Sums

[Paul Hanna]

SeqFans,
      Here are some curious sums I found that I'd like to share with you.

(1) Sum_{n=-oo..+oo}  x^n * (1 - x^n)^n  =  0.

(2) Sum_{n=-oo..+oo, n<>0}  x^n * (1 - x^(n-1))^n / n  =  -log(1-x).

These series are the motivation behind sequence https://oeis.org/A291937 ,
in which I've recorded some related identities.

As an example of (2) at x=2, we may more simply express the doubly-infinite
sum as P + Q, where the infinite series P and Q begin:

P = Sum_{n>=1}  -(-1)^n * 2^n / (n * (2^(n+1) - 1)^n).
Q = Sum_{n>=1} (2^(n-1) - 1)^n / (n * 2^(n^2)).

Explicitly,

P = 2/(1*3) - 4/(2*7^2) + 8/(3*15^3) - 16/(4*31^4) + 32/(5*63^5) -
64/(6*127^6) + 128/(7*255^7) - 256/(8*511^8) + 512/(9*1023^9) -
1024/(10*2047^10) + 2048/(11*4095^11) - 4096/(12*8191^12) +
8192/(13*16383^13) - 16384/(14*32767^14) + 32768/(15*65535^15) +...

Q = 0/(1*2) + 1^2/(2*2^4) + 3^3/(3*2^9) + 7^4/(4*2^16) + 15^5/(5*2^25) +
31^6/(6*2^36) + 63^7/(7*2^49) + 127^8/(8*2^64) + 255^9/(9*2^81) +
511^10/(10*2^100) + 1023^11/(11*2^121) + 2047^12/(12*2^144) +
4095^13/(13*2^169) + 8191^14/(14*2^196) + 16383^15/(15*2^225) +...

where P + Q = log(2)
and
P = 0.6266361387894363397192241172809626592...
Q = 0.0665110417705089696980080041772139088...

Many such logarithmic series can be gleaned from (2) at different values of
x.

I welcome you to explore similar series.
       Paul