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Topic: Taylor Series Error
Replies: 2   Last Post: May 11, 1998 11:39 AM

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John Graham

Posts: 322
Registered: 12/6/04
Taylor Series Error
Posted: May 7, 1998 11:08 AM
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Some AP BC Calculus free-response questions from the past (e. g. 5(c)
from the 1994 exam) seem to have expected students to use the error
term derived from the Alternating Series Test (abs(a sub n + 1)) to
estimate the difference between a partial sum of a Taylor series and
the function that gave rise to it. Mathematically, this seems dubious
to me. The error term from the AST merely gives an estimate of the
difference between a partial sum and the infinite sum of the series,
which says nothing about convergence to the given function. As we
know, there are Taylor series that converge, but not to the function:
The classic example is e^(-1/x^2), expanded about 0.

Is there a problem here?

John Graham
Buckingham Browne & Nichols School
Cambridge, MA 02138

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