# Maclaurin Series for ln(1+x): How-to & Steps

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• 0:00 Maclaurin Series: f(x)…
• 7:57 The Summation Expression
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Lesson Transcript
Instructor: Gerald Lemay

Gerald has taught engineering, math and science and has a doctorate in electrical engineering.

In this lesson, we show how to find the Maclaurin series for a particular function: ln(1 + x). In addition to the steps for finding this series, you will also learn how to determine its region of convergence.

## The Maclaurin Series for f(x) = ln(1 + x)

The general expression for the Maclaurin series is given by the formula:

The formulation for the Maclaurin series is complete when we specify the region of convergence. Let's carefully detail five steps for determining the Maclaurin series of f(x) = ln(1 + x).

#### Step 1: Find Derivatives for f(x)

The derivative of ln(x) is 1/x. Thus, the derivative of ln(1 + x) is 1/(1 + x):

The second derivative of f(x) is the derivative of f '(x):

We write 1/(1 + x) as (1 + x)-1:

The derivative of (1 + x)-1 is (-1)(1 + x)-2 where the exponent, -1, goes in front and the exponent has been reduced by 1 to become -2:

Moving the (1 + x)-2 to the denominator, we get:

Differentiating again will give 2/(1 + x)3:

The next derivative will be -6/(1 + x)4. The next derivative will be 24/(1 + x)5. Looking ahead, it will be definitely useful to have an general expression for the nth derivative. Let's walk through this.

• Note the signs alternate between positive and negative. We express this as (-1)n+1 for n = 1, 2, 3, etc.
• The numerator of the derivative is 1, 1, 2, 6, 24, . . . for n = 1, 2, 3, 4, 5, . . . which agrees with (n - 1)! Note 0! = 1, 1! = 1 and 2! = 2(1) = 2. The exclamation point symbol is called the factorial.
• The denominator is (1 + x) raised to the n.

Thus, the nth derivative is as follows:

#### Step 2: Evaluate These Derivatives and f(x) at x = 0

For f(x) = ln(1 + x), let x = 0. Thus, f(0) = ln(1 + 0) = ln(1) = 0 meaning the first term in this series is 0. The first 4 derivatives evaluated at x = 0.

By the way, the expression for the nth derivative evaluated at x = 0.

We'll use this later when we determine the region of convergence.

#### Step 3: Assemble the Sum of Products

Remember the first line of the general expression for the Maclaurin series?

Let's substitute what we know into this expression.

As already shown, f(0) = 0.

The second term is:

The third term is a little more complicated:

Using the first line as a model, we can deduce the terms which follow. In the fourth term we will see a 3! in the denominator, which evaluates to 3(2)(1) = 6. With the 2 in the numerator gives 2/6, which reduces to 1/3. As you can see, this ultimately equals:

The fifth term will have 4! = 4(3)(2)(1) = 24. The numerator is 6 giving 6/24 = 1/4. As you can see, it ultimately turns into:

Thus, the Maclaurin series for ln(1 + x) is this:

#### Step 4: Find the General Term

We'll need the general term when we explore the region of convergence. From line two of the general expression, we see:

Recall the nth derivative evaluated at x = 0, which is:

Thus, the general term ends up being:

Which is valid for n = 1, 2, 3, etc. Note that n! is n(n-1)(n-2). . . (1) or just n! = n(n-1)!. The (n-1)! terms cancel leaving 1/n.

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