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We can find Maclaurin series for more complicated functions by combining multiple Maclaurin series.
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The Maclaurin series for a composite function βf(g(x))β is
Basically, we replace βxnβ with β[g(x)]n.
We can add or subtract Maclaurin polynomials by adding or subtracting the coefficients of each power of βxβ from the original polynomials.
We can integrate a function by integrating its Maclaurin polynomial term by term.
We can differentiate a Maclaurin series using the power rule on each term.
We can find the first few terms in the Maclaurin Series for a product βf(x)Γg(x)β by multiplying out the first few terms in the Maclaurin Series for βfβ and βg.
We can perform "division" of Maclaurin series by assuming that the result of the division is a series with
where βanββ are the coefficients of the βnβth power term. Then, we multiply βg(x)f(x)ββ and βa0β+a1βx+a2βx2+...β by βg(x). Finally, we solve for the unknown coefficients using the Maclaurin series of βf(x).
Certain limits may be evaluated with LβHΓ΄pital's rule but require difficult or repeated differentiation. Sometimes, expanding a function within the limit using its Maclaurin series and simplifying is an easier method for evaluating these limits.
This kind of simplification often works because we know that after a certain point, every term will tend to β0, allowing us to focus on the few terms that do not.