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