91Ó°ÊÓ

The Taylor series expansion is a fundamental tool in mathematics that allows us to approximate functions using infinite sums of polynomial terms. We start with a function, like \( \cos x \), and express it in the form:\[\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots\]Here, each term in the series is derived from the successive derivatives of \( \cos x \) evaluated at zero, giving us a way to express \( \cos x \) as an infinite series when close to zero.
In the integral we're solving, we take this expansion and adjust it for our needs, which involves substituting it into the function \( \frac{\cos x - 1}{x} \).
This adjustment sets up the series to account for the additional term \( -1 \) in the numerator, leaving us with an expression simpler to integrate right away and is crucial for further steps in our solution.

The method of term by term integration involves integrating each term in a series individually with respect to \( x \).
Given our series from the Taylor series expansion:\[-\frac{x}{2} + \frac{x^3}{24} - \frac{x^5}{720} + \cdots\]We apply the basic integral formula for power functions, \( \int x^n \, dx = \frac{x^{n+1}}{n+1} \).
This transforms each term independently as follows:

• The term \(-\frac{x}{2}\) becomes \(-\frac{x^2}{4}\).
• \(\frac{x^3}{24}\) becomes \(\frac{x^4}{96}\).
• \(-\frac{x^5}{720}\) becomes \(-\frac{x^6}{4320}\).

Term by term integration is particularly useful since each step is straightforward and follows directly from the previous one, eventually forming a series solution.

Infinite series integration is about summing an infinite number of terms, each integrated term by term, to find a solution to an integral.
This method is helpful when direct integration is difficult or impossible.
By beginning with a Taylor expansion and simplifying, as shown in our problem, we can transform complex functions into manageable sums.
Here's how it works in a nutshell:

• Develop a Taylor expansion of the function around a point (usually 0 for simplicity).
• Integrate each term separately using standard integration techniques.
• The result is presented as a sum of all these individual integrated terms, often added by an arbitrary constant \( C \) indicating the indefinite nature.

The final series answer for the problem, \( \int \frac{\cos x - 1}{x} \), was thus expressed as an infinite series:\[-\frac{x^2}{4} + \frac{x^4}{96} - \frac{x^6}{4320} + \cdots + C\]This approach provides an answer even when a function does not possess a straightforward antiderivative.