Educator

Taylor Series and Maclaurin Series

Main definitions:

Definitions: The Taylor Series for a function f (x) around a center value a is the power series

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Here f ⁽ⁿ⁾ (a) represents the n-th derivative of f, with a plugged in.

The Maclaurin Series for f (x) is just the special case of the Taylor Series around the center value a = 0:

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The Taylor polynomial is what you get when you cut off the Taylor Series at the degree k term:

=

Hints and tips:

• In many cases, you do not want to use the formulas above to find the Taylor Series of a function, because the derivatives get too messy. Instead, start with some known Taylor Series for some common function and derive other series from the known series using the following techniques:

• Algebraic manipulations, e.g. multiplying by x.

• Substitutions, e.g. replacing x by 2x or x² .

• Derivatives and integrals.

• Multiplying or dividing two series together.

• You should memorize the Maclaurin Series for e^x , sin x, and cos x at the very least, and probably for 1/(1−x), arctan x, and ln(1 − x) as well.

• Sometimes you cannot find the general pattern for a Taylor Series, especially those that are not centered at a = 0. However, you can still find the first few terms, and this might be enough for computations.

• The Taylor series for a polynomial is just the polynomial itself. A common mistake is to think that the Taylor polynomial T_k (x) has k terms. k refers to the degree, not the number of terms. So, for example, the Taylor polynomial T₄ (x) for f (x) = sin x centered around a = 0 is T₄ (x) = x − x³⁄ 6 , because the term of x⁴ is zero.