Evaluating the sum of the first n terms of an alternating series, given by
, yields an approximation of the limit, L. The error (the difference
between the approximation and the true limit) is called the remainder after n
terms and is denoted by Rn. When an alternating series is first shown to pass the
Alternating Series Test, it’s easy to place an upper bound on this remainder.
Because the terms alternate in sign and become progressively smaller in
magnitude, an alternating series converges on its limit by oscillation, as shown in
Figure N10–1.
Figure N10–1Alternating series error boundBecause carrying out the approximation one more term would once more
carry us beyond L, we see that the error is always less than that next term. Since
|Rn| < an (^) + 1 , the alternating series error bound for an alternating series is the
first term omitted or dropped.
BC ONLY
Example 31 __
The series passes the Alternating Series Test; hence its sum differs
from the sum
by less than , which is the error bound.
Example 32 __
Use the alternating series error bound to determine how many terms must be