Maclaurin Series for ln(1+x)

Deriving the Maclaurin expansion series for ln(1+x) is very easy, as you just need to find the derivatives and plug them into the general formula.

As you can see ln1 = 0

Once you differentiate, you end up with a simple reciprocal.

Differentiating it again simply increases the power as you can see.

In the numerator, you have a factorial sequence occurring, which is great. Once you have a pattern you can do this in your head easily.

The only part that changes is the numerator where there is a factorial sequence building.

This is the pattern of the numbers.

It is a factorial pattern, and algebraically it cancels out with the factorials in the denominator in the Maclaurin general formula.

Here you can see the values substituted into the Maclaurin general form. The factorials cancel out with the numbers at the top to leave a simplified term.

As you can see this is a very simple alternating sequence. Just for fun, see what happens when x = -1 or any value greater than 1. Although it is a simple series, it behaves very strangely.