An alternating series converges as long as thesequenceitself goes is a decreasing sequence (decreasing to 0). On thing that means is that each partial sum lies between the previous two partial sums. Thetruesum of the series must lie between any two partial sums: Since lies between and [tex]\sum_{i=0}^{n+1} (-1)^i a_i[tex], the distance from one of those to the correct value cannot be larger than the difference betwween those two values: the error cannot be larger than .