The first summand has much larger absolute value than the second, so we could solve a simpler, approximate equation
In[4]:= NSolve[16 (1/5)^(2 k + 1)/(2 k + 1) == 10^-102, k, Reals]
Out[4]= {{k -> 71.7807}}
Then compute the terms with k = 71 and k = 72 and see if they are acceptable.
