sum log(base4)2-log(base8)2+log(base 16) 2 - ..

the nth term = log( base (2^(n+1) 2 = 1/(n+1)

sign is alternating

so we get

S = 1/2 - 1/3 + 1/4 - 1/5 + ....

we have from http://www.mathkb.com/Uwe/Forum.aspx/mat…

ln 2= 1 - 1/2 + 1/3 - 1/4 ....

so ln 2 = 1- s

so S = 1 - ln 2