91Ó°ÊÓ

A power series is a way to express a function as an infinite sum of terms. These terms are powers of a variable, often denoted by \(x\). For the function \( \frac{1}{(1+x)^{4}} \), we express it using a power series expansion, which helps us understand the behavior of the function for small values of \(x\). In the binomial series expansion, a power series is used to represent expressions of the form \((1+x)^{n}\), where \(n\) can be any real number. The format of the series is:\[(1 + x)^n = \sum_{k=0}^{\infty} \binom{n}{k} x^k\]This means the function is expanded into an infinite series, where each term includes a power of \(x\). Expanding \((1+x)^{-4}\), as derived from the exercise, we find:

• \(1\)
• - \(4x\)
• \(+ 10x^2\)
• - \(20x^3\)
• and so on...

This step-by-step expansion helps in approximating the function for very small \(x\), which proves beneficial in calculus and analysis. This also allows us to integrate, differentiate, or evaluate functions more easily.

The radius of convergence refers to the range of \(x\)-values for which a power series converges to a particular function. For the binomial series, this is crucial to determine where the series provides an accurate representation of the function.In the context of our expression \( (1+x)^{-4} \), the radius of convergence indicates the distance from the center (usually 0 or another specific point) within which the series will converge. The rule to find this radius for the binomial series \((1 + x)^{-n}\) is simple: the series converges when \(|x| < 1\). Here, this means the power series expansion of \( \frac{1}{(1+x)^4} \) remains valid for \(x\) values between -1 and 1:

• \(-1 < x < 1\)

Beyond this interval, the series is not guaranteed to converge, potentially making the power series unusable for practical calculations. Knowing the radius of convergence is vital to ensure the series provides an accurate depiction of the function.

The generalized binomial coefficient is an extension of the standard binomial coefficients. These coefficients are used when expanding expressions such as \((1+x)^{-n}\). Unlike the standard binomial coefficients that rely only on positive integers, generalized coefficients can handle any real number for the exponent.For a negative exponent as in \((1+x)^{-4}\), the generalized binomial coefficient \(\binom{-n}{k}\) is calculated using the formula:\[\binom{-n}{k} = \frac{(-n)(-n-1)(-n-2)\cdots(-n-k+1)}{k!}\]This allows us to find the coefficient of \(x^k\) in the power series. For example, when \(k=2\):

• We compute \(\binom{-4}{2} = \frac{(-4)(-5)}{2!} = 10\)

This formula lets us systematically determine each term in the series expansion. Mastery of this concept is essential for properly handling and expanding power series where the exponent might not be a simple positive number. It broadens the applicability of the binomial series to a wider array of mathematical functions.