Maclaurin series of the exponential function

In this section we will use the Maclaurin series to find a polynomial approximation to the exponential function, e^x.

The general formula for the Maclaurin series for the function f(x) is:

Where:

• f(0) is the value of the function for x = 0.
• f'(0) is the value of the first derivative function for x = 0.
• f''(0) is the value of the second derivative function for x = 0.
• f'''(0) is the value of the third derivative function for x = 0.
• And so on.

To apply this to the exponential function, we need to calculate those derivatives.

Derivatives the exponential function

The exponential function is unique in that e^x is its own derivative, that is:

This also applies to the second derivative (and therefore to the third derivative and so on):

We also know that any positive number raised to the power zero is one, so e⁰ is one. This means that when x= 0, the value of the exponential function and every nth derivative of the exponential function is equal to one.

This means that:

• f(0) = 1
• f'(0) = 1
• f''(0) = 1
• etc

Maclaurin series of exponential function

Taking the general equation above:

We can replace f(0) and all of its derivatives with 1, giving:

Or using sigma notation (as described here):

A graphical illustration of the Maclaurin expansion

Here is an animation that shows the first 4 terms of the expansion being added in one by one:

One way to gain an intuitive insight into how the Maclaurin expansion works is to look at graphs of the approximation as we add the terms one by one.

Step 1

Taking just the first term of the expansion gives us:

This graph shows the expansion (in yellow) and the exponential function (in black):

This approximation is not very good, it is just a horizontal straight line which is nothing like the exponential function.

However, if we look at the two functions when x = 0 we see that they both have the same value, 1.

The first term of a Maclaurin series ensures that the value of the series equals the value of the function at x = 0

Step 2

Taking the first two terms of the expansion gives us:

Here is the graph:

This approximation is still not very good. However, you will notice that the line now forms a tangent to the curve.

This means that when x = 0, the two functions have the same value and the same slope.

The second term of a Maclaurin series ensures that the slope of the series equals the slope of the function at x = 0

Step 3

Taking the first three terms of the expansion gives us:

Here is the graph:

This approximation is now looking a bit better. The curves are roughly similar for values of x between -0.5 and +0.5.

This is because the approximation now matches the value, slope and second derivative of the curve at x = 0.

The third term of a Maclaurin series ensures that the second derivatives of the series and the function are equal at x = 0

Step 4

Taking the first four terms of the expansion gives us:

Here is the graph:

This approximation is now looking quite convincing. The curves are roughly similar for values of x between -1 and +1.

The approximation now matches the value, slope, and the second and third derivatives of the curve at x = 0.

The fourth term of a Maclaurin series ensures that the third derivatives of the series and the function are equal at x = 0

See also

• Calculating a Maclaurin series
• Maclaurin series of the sine function
• Pi isn't about circles