Calculus 2 : Lagrange Error

Example Questions

Example Question #61 : Polynomial Approximations And Series

Let  be the fifth-degree Taylor polynomial approximation for , centered at .

What is the Lagrange error of the polynomial approximation to ?

Explanation:

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 #1 : Lagrange's Theorem

Which of the following series does not converge?

Explanation:

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.