Show that the series 1/(2log2)+1/(3log3)-1/(4log4)-1/(5log5)-1/(6log6)-1/(7log7)+.... is convergent, the rule of signs being that successive terms with the same sign come in groups of 2, 4, 8, 16... Begin by considering the series 1/2+1/3-1/4-1/5-1/6-1/7+1/8...

What i know is that the sum of 1/logn goes to 0 and decreasing, so that part is fine, but I can't find a bound on the 1/2+1/3-1/4-1/5-1/6-1/7+1/8... to satisfy the Dirichlet Test condition.