Note this is an alternating series so you can use |S-Sn| < a(n+1)
replace x with .4
you get (-1)^(n+1) *.4^(n+1)/(n+1)
|S-Sn| < .4^(n+2)/(n+2) < .001
Use a table to find the smallest n to satisfy this inequality
For the alternating series, if we stop the interation after the summing of the term , the [obvious] error criterion is...
(1)
Computing the 'infinite sum'...
(2)
... the condition is met for n= 5, i.e. five term of (2) are enough and that gives...
(3)
The 'exact' value is...
(4)
Kind regards
The series expansion...
(1)
... converges in genearal for complex values of z if |z|<1. It is interesting the computation of (1) for . Summing 'only' 44 complex terms of (1) we obtain...
... where both real and imaginary part have an error less than ...
Kind regards