Consider the alternating series

which converges by the AST. Find the error for each of the partial sums

(a) *S*_{9}

(b) *S*_{98}

Answer

(a) Look at some partial sums:

The partial sums jump down, then up, with *S*_{9} 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 *S*_{9}.

In symbols,

(b) We have

Since the last term in this partial sum has a negative sign, we jumped down to get to *S*_{98}. To get to *S*_{99} we jump up by , but this is too big a jump. The value we want is within of *S*_{98}. In symbols,