There are three common ways to get new series from old series without calculating all the derivatives and substituting into the Taylor Series general term.

- Substituting into known series
- Differentiating and integrating
- Approaching rational functions as geometric series

This will be the subject of this and my next two posts.

**Substituting**

As you might expect the Taylor/Maclaurin series for the “parent” functions are known. We’ve seen the series for ln(*x*) and sin(*x*) and you can develop or look up series for *e ^{x}*, cos(

*x*) and so on. The power series for compositions involving these series can be found by substituting.

Example 1: To find the Maclaurin series for sin(3*x*), substitute 3*x* for *x* in the sin(*x*) series

Substituting along with some algebra also works, as the next example shows.

Example 2: To find the series for multiply the last answer by *x*^{2}

Example 3: You must be a little careful sometimes. A recent AP Calculus exam asked for the first 4 terms of the Maclaurin series for . Substituting into the sin(*x*) series gives a series centered at but the questions required a series centered at *x* = 0. Almost all students calculated derivatives and used them in the general form of a Maclaurin series. But with a little trigonometry you can substitute after some rewriting:

Now we can substitute into the series for the sin(*x*) and cos(*x*):

As you can see from the last two examples, any sort of “legal” algebra or trigonometry can be used to find a new series from one you already know.

Next post: Differentiating and Integrating

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