# Taylor series (very easy but have a problem)

• FermatPell
In summary, the problem involves finding the series expansion at c=2 of ln(x^2+x-6). By substituting y=x-2, we can simplify the equation to ln(y^2+5y), which can be expanded using the Taylor series of ln(1+x). However, upon checking, it is found that the point x=2 is not a valid point for the expansion.
FermatPell

## Homework Statement

series expansion at $$c=2$$ of $$ln(x^2+x-6)$$

## The Attempt at a Solution

After substituting $$y= x -2$$ we get ln(y^2+5y) = ln(y) + ln(y+5) but I am not kinda sure how to use the taylor series of ln(1+x)...

Hi Fermatpall,

ln(x^2+x−6) = ln((x+3)(x-2)) = ln(x+3) + ln(x-2)

The problem tells us to expand the taylor series around c = 2. The formula we need is:

$$\sum_{n = 0} ^ {\infty} \frac{f^{(n)}(c)}{n!} \, (x - c)^{n}$$

Let's crank out some derivatives then:

$$f'(x) = \frac{1}{x+3} + \frac{1}{x-2}$$
.
.
.

Calculate a few more derivatives and you'll find a pattern. After this is done, throw your result into the formula.

Hmm, I don't think there is a Taylor expansion around x=2, since $\ln(2^2 + 2 -6)=-\infty$.

I like Serena said:
Hmm, I don't think there is a Taylor expansion around x=2, since $\ln(2^2 + 2 -6)=-\infty$.

Good call. I didn't check to see if the point was even valid.

## 1. What is a Taylor series?

A Taylor series is a mathematical representation of a function as an infinite sum of terms, where each term is a derivative of the function evaluated at a specific point.

## 2. Why are Taylor series important?

Taylor series are important because they allow us to approximate complex functions with simpler polynomial functions, making it easier to analyze and solve problems in calculus and physics.

## 3. What is the purpose of using a Taylor series?

The purpose of using a Taylor series is to approximate a function at a specific point, especially when the function is difficult to evaluate directly. This can also help us understand the behavior of a function near a certain point.

## 4. How do you find the terms of a Taylor series?

The terms of a Taylor series can be found by taking the derivatives of the function at the point of interest and evaluating them at that point. The coefficients of each term can be found using the formula: fn(a)/n!, where n is the order of the derivative and a is the point of interest.

## 5. What is the difference between a Taylor series and a Maclaurin series?

A Taylor series is centered around any point a, while a Maclaurin series is centered around 0. In other words, a Maclaurin series is a special case of a Taylor series where a=0. This means that the Maclaurin series only contains non-negative powers of x, making it easier to evaluate and use in certain situations.

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