Consider the alternating series
which converges by the AST. Find the error for each of the partial sums
(a) Look at some partial sums:
The partial sums jump down, then up, with S9 being an up-jump.
To get to the next partial sum we jump down:
and in so doing, we overshoot the line:
This means when we jumped down by , it was too much. The real sum of the series, L, must be within of S9.
(b) We have
Since the last term in this partial sum has a negative sign, we jumped down to get to S98. To get to S99 we jump up by , but this is too big a jump. The value we want is within of S98. In symbols,