Which of the following is the Taylor series of ln(1−2x) about x=0 up to and including the terms of order three?
What is order three?
In mathematics, the "order" of a power series refers to the highest power of the variable in the series. For example, the order of a series with terms up to the third power would be three. Therefore, when it says "terms of order three," it means we need to find the Taylor series expansion of ln(1-2x) up to and including terms with x raised to the power of three.
To find the Taylor series expansion of ln(1-2x) about x=0, we can use the following steps:
1. Calculate the derivatives of the function ln(1-2x) up to the desired order. In this case, we need derivatives up to order three. The derivatives are as follows:
- First derivative: (-2)/(1-2x)
- Second derivative: 4/(1-2x)^2
- Third derivative: -16/(1-2x)^3
2. Evaluate each derivative at x=0. Substituting x=0 into the derivatives, we get:
- First derivative at x=0: -2
- Second derivative at x=0: 4
- Third derivative at x=0: -16
3. Use the values obtained in step 2 to write the Taylor series expansion. The Taylor series expansion of ln(1-2x) about x=0 up to and including terms of order three is given by:
ln(1-2x) ≈ -2x + 2x^2 - (8/3)x^3
Now we have the Taylor series expansion of ln(1-2x) about x=0 up to and including the terms of order three.