# Range of Validity for Maclaurin Expansion of ln(1-x)

• josephcollins
In summary, the Maclaurin expansion of ln(1-x) is an infinite series that represents the natural logarithm function for values of x near 0. Its range of validity is limited to -1 and 1, as the series only converges for values within this range. This is due to the radius of convergence for ln(1-x) being 1. Using the Maclaurin expansion for values outside the range of validity can result in inaccurate results. The range of validity for other Maclaurin expansions can be determined by finding the radius of convergence using mathematical tests such as the ratio test or the root test.
josephcollins
Hi ppl. May I ask how you obtain the range of convergence for the maclaurin expansion of ln(1-x) and taylor's series, maclaurin's in general? Thanks, Joe

A power series is uniformly convergent everywhere inside its radius of convergence, so you just need to find that. The ratio and root tests are good bets.

Hi Joe,

The Maclaurin expansion of ln(1-x) is given by:

ln(1-x) = -x - (x^2)/2 - (x^3)/3 - (x^4)/4 - ...

This expansion is valid for values of x within the range of convergence, which is -1 < x < 1. This means that the series will converge to the exact value of ln(1-x) for any value of x within this range.

To determine the range of convergence for this series, we can use the ratio test. This test states that if the limit of the absolute value of the ratio of consecutive terms is less than 1, then the series will converge.

In this case, we have:

lim(n->∞) |(x^n)/(n)| = lim(n->∞) |x^n| = |x|

Since we want this limit to be less than 1, we need to have |x| < 1, which gives us the range of convergence as -1 < x < 1.

This same process can be applied to find the range of convergence for any Maclaurin or Taylor series. It is important to note that the range of convergence can vary for different series, so it is necessary to check each individually.

I hope this helps answer your question. Let me know if you have any further questions.

## What is the Maclaurin expansion of ln(1-x)?

The Maclaurin expansion of ln(1-x) is an infinite series that represents the natural logarithm function for values of x near 0.

## What is the range of validity for the Maclaurin expansion of ln(1-x)?

The range of validity for the Maclaurin expansion of ln(1-x) is between -1 and 1, since the series only converges for values of x within this range.

## Why is the range of validity for the Maclaurin expansion of ln(1-x) limited to -1 and 1?

This is because the Maclaurin expansion is based on the Taylor series, which only converges for values within the radius of convergence. For ln(1-x), this radius of convergence is 1.

## Can the Maclaurin expansion of ln(1-x) be used for values outside the range of validity?

No, the Maclaurin expansion of ln(1-x) is only valid for values between -1 and 1. Using it for values outside this range can lead to inaccurate results.

## How can I determine the range of validity for other Maclaurin expansions?

The range of validity for Maclaurin expansions can be determined by finding the radius of convergence using various mathematical tests, such as the ratio test or the root test. Consult a calculus textbook or online resources for more information on these tests.

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