11: AP Calculus Reference_BC_Series & Tests of Convergence and Divergence

Tests of Convergence and Divergence

If one test is inconclusive, try a different test. There is no single test that must be used on a given series.

Taylor Polynomial

What is the general form of a Taylor polynomial for f(x) centered at x=c?

Taylor Series vs. Taylor Polynomial

What is the difference between a Taylor series and a Taylor polynomial?

• Taylor Polynomial: Finite sum up to the n-th term.

• Taylor Series: Infinite sum (if it converges to f(x)).

Taylor Remainder (Error Term)

How is the error Rn(x) of a Taylor polynomial defined?

• Lagrange Error: Use when you need a worst-case error estimate for Taylor approximations.

• Alternating Series: For series like ∑(−1)ⁿa_n, the error is always less than the next term.

Maclaurin Series