91Ó°ÊÓ

A geometric series is a sum of terms where each term is a constant multiple of the previous one. In simple cases, it looks like a sequence such as 1, \, r, \, r^2, \, r^3, \ldots, where \(r\) is the common ratio. Geometric series are important because they can often be summed up into a simple expression, allowing complex functions to be expressed in series form. In the context of the problem, the Taylor series for \(-1/(1+x)\) is a geometric series. The common ratio here is \(-x\), starting with \(-1\):

• First term: \(-1\)
• Second term: \(x\)
• Third term: \(-x^2\)
• And so on...

This highlights the power of geometric series to represent functions cleanly, making it easier to derive new series through techniques like differentiation.

Differentiation is the process of finding the derivative of a function, which represents the rate at which the function changes. For a series, differentiation can be applied term by term. In the solution provided, the Taylor series of \(-1/(1+x)\) was differentiated to generate a new series for \(-1/(1+x)^2\). By differentiating term by term:\(-1 + x - x^2 + x^3 - x^4 + \ldots\), we obtained:

• Derivative of \(-1\) is 0.
• Derivative of \(x\) is 1.
• Derivative of \(-x^2\) is \(-2x\).
• For \(x^n\), the derivative is \(nx^{n-1}\).

This resulted in a new series: \(-1 + 2x - 3x^2 + 4x^3 - \ldots\). Differentiation was pivotal to transforming the original series into something more specific for our function \(1/(1+x)^2\).

Series expansion is a method of expressing a function as an infinite sum of terms. For many functions, especially those not easily simplified, a series expansion like the Taylor series is useful. The problem at hand required us to obtain the Taylor series for \(1/(1+x)^2\) by transforming another known series. To "expand" a function:

• Start with a known series, like the geometric series.
• Use operations such as differentiation to derive a new series.
• Write down the general form of terms, often involving powers and factorials.

Here, the series for \(-1/(1+x)\) was used as a base, expanded via differentiation, and adjusted to achieve the desired expansion: \(\sum_{n=0}^{\infty} (-1)^n (n+1)x^n\). By negating the differentiated series appropriately, we successfully adapted it to suit \(1/(1+x)^2\). This expansion gives a clear, infinite series representation of the function, perfect for approximations and computational evaluations.