# Solve Maclaurin Series for f(x)= 1/(1+x+X2)

• Crazy Gnome
In summary, the Maclaurin series for f(x) = 1/(1+x+x^2) can be found by dividing 1 by the Maclaurin series of 1+x+x^2, or by finding the derivatives of f(x) and using the formula, which results in 1-x+x^3. The second method is accurate, while the first method would also be accurate if polynomial division is used correctly.
Crazy Gnome

## Homework Statement

Find the Maclaurin series for f(x)= 1/ (1+x+X2)

## The Attempt at a Solution

I think the book says I can just divide 1 by the Maclaurin series of (1+x+X2). And when i do this the original function is the answer (which makes sense).

But when I do it my finding the derivatives of f(x) and then using the formula I get f(x) = 1-x+x3.

So I was wondering if
1) Which way is correct
2) is the second one accurate.

Thanks
-James

The second way is accurate. The first way would be as well if you do what they want you to. You are supposed to divide 1 by 1+x+x^2 using polynomial division. The result is NOT 1+x+x^2.

## 1. What is a Maclaurin series?

A Maclaurin series is a type of mathematical series that represents a function as an infinite sum of terms. It is named after Scottish mathematician Colin Maclaurin and is a special case of Taylor series where the series is centered at x=0.

## 2. How do you solve for a Maclaurin series?

To solve for a Maclaurin series, you need to use the general formula for Maclaurin series which is f(x) = f(0) + f'(0)x + f''(0)x^2/2! + f'''(0)x^3/3! + ... + f^n(0)x^n/n!, where f(0) is the value of the function at x=0 and f^n(0) represents the nth derivative of the function at x=0.

## 3. Why is it useful to find the Maclaurin series of a function?

Finding the Maclaurin series of a function can be useful in many ways. It can help us to approximate the value of a function at a specific point, to evaluate integrals and derivatives of the function, and to understand the behavior of the function near the point of expansion.

## 4. How do you find the Maclaurin series for f(x)= 1/(1+x+X2)?

To find the Maclaurin series for f(x)= 1/(1+x+X2), we first need to rewrite the function as a geometric series by factoring out an x. This gives us f(x)= 1/(1-x^2). Then, we can use the formula for the geometric series to find the Maclaurin series, which is f(x)= 1 + x^2 + x^4 + x^6 + ... = Σ(x^n)^2.

## 5. What is the significance of the Maclaurin series for f(x)= 1/(1+x+X2)?

The Maclaurin series for f(x)= 1/(1+x+X2) is significant because it represents a rational function as an infinite polynomial. This allows us to easily evaluate the function and its derivatives at any point, making it a useful tool in many mathematical applications.

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