Out[542]= Sum[((-1)^n*Floor[(EulerGamma + Log[n])/Log[2]])/n, {n,∞}] == 0 (%544 = %542 with terms grouped pairwise, comforting NSum.) Testing: In[547]:= tim[Out[544][[1]] /. Sum[a_, b_] :> NSum[a, b, NSumTerms -> 9999]] During evaluation of In[547]:= NIntegrate::ncvb (NSum remains uncomfortable.) During evaluation of In[547]:= 212.330211,0 Out[547]= -0.000047302360632835656 In[550]:= tim[Out[544][[1]] /. Sum[a_, b_] :> NSum[a, b, NSumTerms -> 9999999]] During evaluation of In[550]:= NIntegrate::ncvb During evaluation of In[550]:= 5.096415,0 Out[550]= -5.549175287798655*^-9 (212 sec for 9999 terms, 5 sec for 10^7 terms.) The 1st 16 terms: In[553]:= %542[[1]]/.{∞->16,Sum->Table} Out[553]= {0,1/2,-2/3,1/2,-3/5,1/2,-3/7,3/8,-4/9,2/5,-4/11,1/3,-4/13,2/7,-4/15,1/4} Repeatedly adding pairwise. In[556]:= Plus@@#&/@Partition[%553,2] Out[556]= {1/2,-1/6,-1/10,-3/56,-2/45,-1/33,-2/91,-1/60} In[557]:= Plus@@#&/@Partition[%,2] Out[557]= {1/3,-43/280,-37/495,-211/5460} In[558]:= Plus@@#&/@Partition[%,2] Out[558]= {151/840,-20431/180180} In[559]:= Plus@@#&/@Partition[%,2] Out[559]= {23917/360360} --rwg