91Ó°ÊÓ

A geometric series is one of the most fundamental series types in mathematics. It arises when each term is a constant multiple of the previous one. Let's consider the series for \( \frac{1}{1+x} \), a classic example of a geometric series. This can be expressed as an infinite sum:

• \( \sum_{n=0}^{\infty} (-1)^n x^n \)

This formula is valid when \(|x| < 1\), ensuring convergence. Here, each term involves raising \( x \) to a higher power, but it's multiplied by \((-1)^n\) which alternates the signs of the terms.
Geometric series are particularly powerful because they can be manipulated easily, often allowing us to find other related series. In this exercise, we're using the geometric series as a starting point to find the Taylor series for another function through differentiation.

Differentiation is a key tool in calculus, allowing us to find the derivative of functions, which represents the rate of change. In the context of the series, we use differentiation to find the derivative of each term in a series separately.
When we differentiate the series \( \frac{1}{1+x} \) term by term, we rewrite it as:

• \( \sum_{n=1}^{\infty} (-1)^n n x^{n-1} \)

This results in the series for \( \frac{-1}{(1+x)^2} \). Note how each term's exponent decreases by one as we differentiate, and each coefficient is multiplied by its original exponent \( n \).
This process demonstrates how differentiation can transform a simple geometric series into another, more complex series, simply by systematically applying differentiation to each term.

Series expansion is the method of expressing a function as an infinite sum of terms. These terms are usually powers or multiples of a variable, allowing us to approximate functions or perform calculations more easily.
In this exercise, after differentiating to obtain \( \frac{-1}{(1+x)^2} \), we multiply the entire series by \(-1\) to align it with the needed function \( \frac{1}{(1+x)^2} \). This step results in:

• \( \sum_{n=1}^{\infty} (-1)^{n+1} n x^{n-1} \)

Finally, we re-index the series. Re-indexing helps express the series in a clearer form by shifting indices, useful in ensuring the starting index matches our interpretation. By setting \( m = n - 1 \), the re-indexed series becomes:

• \( \sum_{m=0}^{\infty} (-1)^{m} (m+1) x^m \)

Here, re-indexing provides a clean, clear way to write the Taylor series expansion, opening the door to numerous applications in approximation and analysis.