Maclaurin Series ln x

The Maclaurin series for ln x does not exist because the derivative of ln x is 1/x and therefore f(0) = 1/0, which is undefined. All the derivatives will be undefined in this way because you would be dividing by zero. If you really need Maclaurin to express natural logs, then you have to shift the curve by 1 to the left. Hence you can have a Maclaurin series for ln(1+x) instead.

Another option is to use the Taylor series centred around a = 1. The general formula above shows the Taylor series, which is very similar looking to the Maclaurin series, and you use it in exactly the same way. You have to find the derivatives of the function and find their values by setting a = 1.

Since a = 1, the first thing you might like to do is to replace it by 1 in the general formula above. All you have to do now is to find the values of the derivatives and substitute them into the equation above. The derivative values are with a = 1 to give a numerical figure. The steps below show how to find the derivatives.

The first one is easy, and as you can see we simply substitute a = 1, and therefore ln 1 = 0.

If a = 1, then the result is 1 for the first derivative.

If a = 1 then the result is -1 for the second derivative.

This is the third derivative.

This is the fourth derivative.

Now all you have to do is to substitute the values into the Taylor series.

This is the Taylor series for ln x centred at a = 1.