Find Maclaurin Series in Easy Steps

Calculus > Find Maclaurin Series

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How to Find Maclaurin series: Overview

Maclaurin and Taylor series help to approximate functions with a series of polynomials. In other words, you’re creating a function with lots of other smaller functions. You can create the number 10 from smaller numbers: 1 + 2 + 3 + 4. In the same way, you can piece together simple polynomials to create a more complicated function. Why would you want to di this? In many cases, you know what a function looks like on a graph; but you don’t know the equation for it. The Maclaurin series can create a duplicate (a doppelgänger, if you like). In other words, it’s so close to the thing you won’t be able to tell the difference.

A Maclaurin series is a special case of a Taylor series, where “a” is centered around x=0. The series are named after Scottish mathematician Colin Maclaurin. While you can calculate Maclaurin series using calculus, many series for common functions have already been found. For example, the following table shows the Maclaurin series for five common functions, along with the sigma notation for the expansion.

If you don’t have one of the most common functions, the Maclaurin series is fairly easy (although somewhat tedious) to calculate.

How to Find Maclaurin series: Steps

Sample problem: Find the Maclaurin series for e5x.

Step 1: Calculate the first few derivatives for the function until you can see a clear pattern. Usually you’ll only need to calculate four or five:

 f(x)=e5x f'(x)=5e5x f”(x)=52e5x f”'(x)=53e5x

Step 2: Fill in the derivatives you calculated in step 1 with 0 as the input. This gives you the values you need to insert into the Maclaurin series:

 f(x)=e5(0) = 1 f'(x)=5e5(0) = 5 f”(x)=52e5(0) = 52 f”'(x)=53e5(0) = 53

Step 3: Fill in the Maclaurin formula with the values you calculated in Step 2:

Step 4: (Optional): Rewrite using sigma notation:

That’s how to find Maclaurin series!

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