Introduction to Maclaurin Series for (x^2(1 x^3)^{-2})

The Maclaurin series is a powerful tool in calculus and mathematical analysis, allowing us to express functions as an infinite sum of polynomial terms. In this article, we will explore the Maclaurin series representation for the function (f(x) x^2(1 x^3)^{-2}). This involves finding the generalized binomial coefficient and understanding the series expansion.

The Maclaurin Series Expanding (f(x) x^2(1 x^3)^{-2})

Let's start with the Maclaurin series expansion for (f(x) x^2(1 x^3)^{-2}). The general form of the Maclaurin series for a function (f(x)) is given by:

[f(x) sum_{n0}^{infty} frac{f^{(n)}(0)}{n!} x^n]

However, for (f(x) x^2(1 x^3)^{-2}), we will use the binomial series expansion for ((1 x^3)^{-2}) to find the series for (f(x)).

Binomial Series Expansion for ((1 x^3)^{-2})

The binomial series expansion for ((1 x)^k) where (k) is any real number is given by:

[(1 x)^k sum_{n0}^{infty} binom{k}{n} x^n]

Here, the generalized binomial coefficient (binom{k}{n}) is defined as:

[binom{k}{n} frac{(k)(k-1)(k-2)cdots(k-n 1)}{n!}]

For our function, we need the expansion of ((1 x^3)^{-2}). Setting (k -2) and (x) to (x^3), we get:

[(1 x^3)^{-2} sum_{n0}^{infty} binom{-2}{n} (x^3)^n]

Calculating the generalized binomial coefficient (binom{-2}{n}), we have:

[binom{-2}{n} frac{(-2)(-2-1)(-2-2)cdots(-2-(n-1))}{n!}]

This expression simplifies as follows for the first few terms:

[binom{-2}{0} 1, quad binom{-2}{1} -2, quad binom{-2}{2} frac{(-2)(-3)}{2!} 3, quad text{and so on}.]

Combining the Series for (f(x) x^2(1 x^3)^{-2})

Now, we need to combine the series expansions to get the Maclaurin series for (f(x) x^2(1 x^3)^{-2}). Using the series expansion for ((1 x^3)^{-2}), we can write:

[x^2(1 x^3)^{-2} x^2 left( sum_{n0}^{infty} binom{-2}{n} (x^3)^n right)]

Expanding this, we get:

[x^2(1 x^3)^{-2} x^2 sum_{n0}^{infty} binom{-2}{n} x^{3n} sum_{n0}^{infty} binom{-2}{n} x^{3n 2}]

Therefore, the Maclaurin series for (x^2(1 x^3)^{-2}) is:

[x^2(1 x^3)^{-2} x^2 (-2)x^5 3x^8 cdots binom{-2}{n} x^{3n 2}]

Conclusion

In summary, we have derived the Maclaurin series for (f(x) x^2(1 x^3)^{-2}) using the binomial series expansion. This results in an infinite series representation that allows us to analyze and understand the behavior of the function over a small interval around (x 0).

Key Takeaways:

The Maclaurin series is a powerful method to express functions as an infinite polynomial series. The generalized binomial coefficient is crucial in expanding functions with non-integer exponents. Combining the series expansions properly is key in obtaining the desired series form.

This article provides a clear and detailed understanding of the Maclaurin series for (x^2(1 x^3)^{-2}), facilitating your ability to apply similar techniques to other functions.