11: AP Calculus Reference_BC_Series & Tests of Convergence and Divergence

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5 Terms

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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.

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Taylor Polynomial

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

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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)).

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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)nan, the error is always less than the next term.

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Maclaurin Series