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The Pochhammer symbol, denoted as \( (a)_n \), is an essential function in mathematics, particularly when dealing with series and factorial-related expressions. It represents the rising factorial power of a number and is defined recursively by \( (a)_0 = 1 \) and \( (a)_{n+1} = (a + n) \cdot (a)_n \). In simple terms, for a non-negative integer n, the Pochhammer symbol is the product of n consecutive numbers starting at a.

For instance, if we take \( a = \frac{1}{2} \) and \( n = 4 \) as an example:
\( \left(\frac{1}{2}\right)_4 = \frac{1}{2} \cdot \frac{3}{2} \cdot \frac{5}{2} \cdot \frac{7}{2} \).

This concept is a fundamental block in understanding hypergeometric functions, as they are expanded in series where each term involves Pochhammer symbols of the parameters of the function. When trying to improve comprehension of step-by-step solutions involving hypergeometric functions, recognizing and working with the Pochhammer symbol is crucial.

Series expansion is a powerful mathematical tool which allows us to express functions as a sum of terms, often making complex functions easier to work with. One common type of series expansion is the power series, where functions are written as an infinite sum of powers of a variable.

A typical power series has the form \( F(x) = \sum_{n=0}^{\infty} a_n \cdot x^n \), where each \( a_n \) is a coefficient that depends on n. Series expansions are not only used in calculus and analysis, but also in more applied fields like physics and engineering, where they simplify problem-solving.

For example, in the step-by-step solution, the hypergeometric function is represented as a sum where each term is a product of Pochhammer symbols and powers of \( -x^2 \). By improving your understanding of series expansion, you can deepen your knowledge on how functions are reconstructed and how they behave as the variable approaches different values.

The inverse tangent function, commonly referred to as \( \arctan \), is one of the inverse trigonometric functions. It is used to determine an angle given the ratio of the opposite side to the adjacent side of a right-angled triangle. Its series expansion plays a vital role in calculus.

The series for \( \arctan x \) can be expressed as \(\arctan x = \sum_{n=0}^{\text{\infty}} \frac{(-1)^n}{2n+1}x^{2n+1}\), which is seen in the step-by-step solution of the exercise.

Understanding \( \arctan \) is important not only in trigonometry but also in complex analysis, signal processing, and other areas of mathematics where angles in the unit circle or periodicity are involved. Mastery over the inverse tangent function and its series expansion will help students in solving integrals, evaluating limits, and understanding convergence behavior of series.