The Alternating Series Error Estimate

[xstetsonx]

summation from 1 to infinite: (5n)/(n^2+1)
What is the maximum amount of error of s4 as compared to the infinite series?

error=S-Sn<bn+1



I got S4=-14/17
and S5=-25/26
but how you suppose to get S? i though you can't tell what a alternating series converge to

 

[HallsofIvy]

The key point about an "alternating series", with decreasing terms, is that the sum up to $a_n$ lies between the sum up to $a_{n-1}$ and the sum up to $a_{n-2}$. And, therefore, the actual sum lies between any two consective partial sums. That is, the entire sum lies between $S_4$ and $S_3$ so the error cannot be larger than the distance between those two.