Question

Find how many terms are required to approximate the sum of the series from 1 to infinity of (-1)^(n+1)/(n^3) with an error of less than 0.001.

I think 11 terms are required.

Answer 1

You know that the ramainder after n terms (Rn) obeys

|Rn| < a_n+1

So, we want
1/(n+1)^3 < 1/1000
1000 < (n+1)^3
10 < n+1
9 < n

So, 10 terms are needed.
Check: S10 = 0.90111
S = 0.90154

S-S10 = 0.00043 < 0.001

However, we need 0.90054 < Sn < 0.90254
and it turns out that only 8 terms are needed.
S7 = 0.90269
S8 = 0.90074