approaching 0, so the error committed by using the first two terms is less than
, then the given approximation formula will yield accuracy to three
decimal places. We therefore require that |x|^3 < 0.0015 or that |x| < 0.114.
C5. Taylor’s Formula with Remainder; Lagrange Error Bound
When we approximate a function using a Taylor polynomial, it is important to
know how large the remainder (error) may be. If at the desired value of x the
Taylor series is alternating, this issue is easily resolved: the first omitted term
serves as an upper bound on the error. However, when the approximation
involves a nonnegative Taylor series, placing an upper bound on the error is
more difficult. This issue is resolved by the Lagrange remainder.
TAYLOR’S THEOREM. If a function f and its first (n + 1) derivatives are
continuous on the interval |x − a| < r, then for each x in this interval
where
and c is some number between a and x. Rn(x) is called the Lagrange remainder.
Note that the equation above expresses f(x) as the sum of the Taylor
polynomial Pn(x) and the error that results when that polynomial is used as an
approximation for f(x).
Lagrange error boundWhen we truncate a series after the (n + 1)st term, we can compute the error
bound Rn, according to Lagrange, if we know what to substitute for c. In practice
we find, not Rn exactly, but only an upper bound for it by assigning to c the value
between a and x that determines the largest possible value of Rn. Hence:
the Lagrange error bound.
BC ONLYExample 48 __
Estimate the error in using the Maclaurin series generated by ex to approximate