# Alternating Series Test

• Feb 16th 2009, 01:27 PM
nikie1o2
Alternating Series Test
Been doing Alternating Series tests but not sure how to figure out the absolute error of the series...

(-1) ^k+1 / K ; n=7

Any help or suggestions? I know i have to use the formula :
l S- Sn l lessthan/equal to An+1

Thank you for your time & help!
• Feb 16th 2009, 01:57 PM
HallsofIvy
An alternating series converges as long as the sequence itself goes is a decreasing sequence (decreasing to 0). On thing that means is that each partial sum lies between the previous two partial sums. The true sum of the series must lie between any two partial sums: Since $\displaystyle \sum_{i=0}^\infty (-1)^i a_i$ lies between $\displaystyle \sum_{i=0}^n (-1)^i a_i$ 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 $\displaystyle \left|\sum_{i= 0}^{n+1}(-1)^ia_i- \sum_{i=0}^{n}(-1)^ia_i\right|= \left| a_i\right|$.