As some of you already figured out, something is awry with my use and interpretation of

Table[Limit[etaMM[x, 30 + n]/etaMM[x + n, 30 + n], x -> Infinity] - 
  N[(-3/Log[3])^n, 30], {n, 1, 10}]

{-1.37235287940800822169041524488*10^40, 
 2.79444910535423129879510807704*10^41, \
-7.63085157751707944108437643731*10^41, 
 2.08377013152699855184232233095*10^42, \
-4.24307018055104941251691654111*10^43, 
 1.15866267590045475709082282171*10^44, \
-3.16398065409903564894573215540*10^44, 
 6.44264535810872405469659678725*10^45, \
-1.75930455845876064949571024220*10^46, \
-1.31188157413621969593288955467*10^47}

because, to the contrary, I can enter and get the following.

In[13]:= Limit[etaMM[x, 30 + 2]/etaMM[x + 2, 30 + 2], x -> Infinity] -
  N[(-3/Log[3])^2, 30]

Out[13]= 0.*10^-28

In[11]:= Limit[etaMM[x, 30]/etaMM[x + 3, 30], x -> Infinity] - 
 N[(-3/Log[3])^3, 30]

Out[11]= 0.*10^-26

In[16]:= Limit[etaMM[x, 30]/etaMM[x + 4, 30], x -> Infinity] - 
 N[(-3/Log[3])^4, 30]

Out[16]= 0.*10^-25

It looks like I've got n doing double work; it's global in etaMM and the table both. This works, though:

In[58]:= Table[
 Limit[etaMM[x, 30 + u]/etaMM[x + u, 30 + u], x -> Infinity] - 
  N[(-3/Log[3])^u, 30], {u, 1, 10}]

Out[58]= {0.*10^-26, 0.*10^-28, 0.*10^-29, 0.*10^-29, 0.*10^-28, 
 0.*10^-28, 0.*10^-27, 0.*10^-27, 0.*10^-27, 0.*10^-26}