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Example Question #1 : Lagrange Error Bound For Taylor Polynomials
Let be the fifth-degree Taylor polynomial approximation for , centered at .
What is the Lagrange error of the polynomial approximation to ?
The fifth degree Taylor polynomial approximating centered at is:
The Lagrange error is the absolute value of the next term in the sequence, which is equal to .
We need only evaluate this at and thus we obtain
Example Question #3 : Alternating Series With Error Bound
Which of the following series does not converge?
We can show that the series diverges using the ratio test.
will dominate over since it's a higher order term. Clearly, L will not be less than, which is necessary for absolute convergence.
Alternatively, it's clear that is much greater than , and thus having in the numerator will make the series diverge by the limit test (since the terms clearly don't converge to zero).
The other series will converge by alternating series test, ratio test, geometric series, and comparison tests.