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Sigma notation is a powerful tool for representing the sum of a sequence of terms in a compact form. When you see a summation symbol, \(\sum_{n=a}^{b} expression \), it indicates that you should evaluate the **expression** for each integer value of \(n\) from \(a\) to \(b\) and then find the sum of these values.

• \(n\) is the index of summation.
• \(a\) is the lower limit where summation starts.
• \(b\) is the upper limit of the summation.

In the context of a Taylor series, sigma notation allows us to succinctly express the infinite sum of terms that arise when expanding a function about a particular point. For instance, for the natural logarithm expansion around \(x_0 = 1\), the sigma notation form begins from \(n=1\) because the first derivative of \(\ln x\) at that point is zero, simplifying our sum. This transforms the infinite series of terms into a formula that is easier to analyze and compute.

Derivative evaluation involves calculating the derivative of a function at a certain point, which is crucial in constructing Taylor series. For a given function like \(f(x) = \ln x\), derivatives help derive the closer approximation of the function as a power series. Let's look at the process step-by-step:

• First Derivative: The first derivative of \(\ln x\) is \(f'(x) = \frac{1}{x}\).
• Second Derivative: Following the first, the second derivative is \(f''(x) = -\frac{1}{x^2}\).
• Higher-order Derivatives: This sequence continues as \(f'''(x) = \frac{2}{x^3}\), showing a pattern of alternating signs and increasing powers of \(x\).

Once the derivatives are found, the next step is to evaluate them at the point \(x_0 = 1\). This provides the coefficients needed for the Taylor expansion:

• \(f(1) = 0\), because \(\ln 1 = 0\).
• \(f'(1) = 1\), as substituting \(x = 1\) in \(\frac{1}{x}\) gives 1.
• \(f''(1) = -1\), and so on.

These evaluated derivatives are crucial because they determine the series terms' coefficients, reflecting the function's behavior near the expansion point.

The natural logarithm expansion is a classic example of an application of Taylor series. By expanding the natural logarithm function \(\ln x\) around a point, we can approximate the function in the form of an infinite series. Generally, this is how Taylor expansions work:

• The base function \(\ln x\) is to be expanded around \(x_0 = 1\).
• Using Taylor's formula, this function is expressed as \( \sum_{n=0}^{\infty} \frac{f^{(n)}(x_0)}{n!}(x-x_0)^n \).

Given that \(f^{(n)}(x_0)\) denotes the \(n\)-th derivative evaluated at \(x_0\), our task is to place these into the sum. Since the logarithm lacks a straightforward pattern at \(x_0=1\), only terms from \(n=1\) onwards contribute:
- The series becomes \(\ln x = \sum_{n=1}^{\infty} \frac{(-1)^{n+1}(n-1)!}{n!}(x-1)^n \), reflecting the characteristic alternating sign and factorial denominator.
This expansion is significant for approximating \(\ln x\) near \(x_0 = 1\), capturing its behavior efficiently and elegantly through calculus.