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(3x), substitute 3x 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 x²

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