# Finding a Maclaurin series for ln(x)

• bobber205
In summary, the conversation discusses finding a Maclaurin power series for ln(x) and determining the number of times it needs to be run to get an accurate answer for ln(1.5). It is suggested to find the Maclaurin series for ln(1 + x) and substitute 0.5 instead of 1.5. However, it is recommended to clarify with the instructor.
bobber205
Since ln(0) doesn't exist, this question is futile right?

I am tasked with finding a Maclaurin powerseries for ln(x) and to find out how many times I have to run that series to get a accurate answer for ln(1.5).

What should I do? Should I find the taylor series for ln(1.5) for should I find the Maclaurin for ln(1 + x) and find out what .5 is instead of 1.5.

Thanks.

Should I find the taylor series for ln(1.5) for should I find the Maclaurin for ln(1 + x) and find out what .5 is instead of 1.5.

Seems like the better choice (Maclaurin for $ln(1 + x)$ or Taylor of $ln(x)$ about $x=1$) - this may be what was intended all along. However, you should clarify with instructor.

## 1. What is the general formula for the Maclaurin series of ln(x)?

The general formula for the Maclaurin series of ln(x) is ln(x) = (x-1) - (x-1)^2/2 + (x-1)^3/3 - (x-1)^4/4 + ...

## 2. How do you find the Maclaurin series for ln(x)?

To find the Maclaurin series for ln(x), we can use the formula ln(x) = (x-1) - (x-1)^2/2 + (x-1)^3/3 - (x-1)^4/4 + ... and plug in the value of x into the formula. Then, we can simplify the terms and continue the pattern of the series to find more terms.

## 3. What is the error term in the Maclaurin series for ln(x)?

The error term in the Maclaurin series for ln(x) is given by Rn(x) = (x-1)^n+1/(n+1)x^(n+1). This term represents how much the Maclaurin series differs from the actual value of ln(x).

## 4. How many terms do you need to include in the Maclaurin series to approximate ln(x) with an error less than a given value?

To approximate ln(x) with an error less than a given value, we need to include enough terms in the Maclaurin series so that the absolute value of the error term Rn(x) is less than the given value. The number of terms needed will depend on the value of x and the desired error.

## 5. Can the Maclaurin series for ln(x) be used to find the value of ln(x) for negative values of x?

No, the Maclaurin series for ln(x) can only be used to find the value of ln(x) for positive values of x. For negative values of x, we can use the Taylor series expansion of ln(1-x) to find the value of ln(x).

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