\(\sum_{n=1}^{\infty} \frac{n !}{(n+1) !}\)

The harmonic series is a famous infinite series that plays a crucial role in mathematics. It's defined as the sum of reciprocals of all positive integers. Mathematically, it can be represented as:
\[ \text{Harmonic Series} = \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \text{...} \] In simpler terms, we keep adding fractions where the numerator is always 1, and the denominator increases by one each time. The harmonic series does not start from 1 in all cases. Sometimes, it may start from any integer n. For example, in this exercise, it starts from 2:
\[ \text{Modified Harmonic Series} = \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \text{...} \] Understanding the harmonic series is essential. It shows up in many areas of mathematics and physics. It has an interesting property: even though individual terms get smaller, the sum grows indefinitely. This phenomenon is known as divergence.

Factorials are a key concept in many areas of mathematics, notably in combinatorics and calculus. The notation n! (read as 'n factorial') represents the product of all positive integers less than or equal to n. For instance, 5! = 5 × 4 × 3 × 2 × 1 = 120. In this exercise, we simplify a fraction involving factorials:
The original form is \[ \frac{n !}{(n+1) !} \] We know that \[ (n+1)! = (n+1) \times n! \] Therefore, we can simplify the fraction as:
\[ \frac{n !}{(n+1) !} = \frac{n !}{(n+1) \times n !} = \frac{1}{n+1} \] This shows us that we can break down complex factorial expressions into much simpler parts. Simplifying factorials often helps in identifying patterns or recognizing well-known series, as seen here with the harmonic series.

In mathematics, whether a series converges or diverges is a fundamental aspect to understand. Convergence means that as we add more terms in the series, the sum approaches a specific value. Divergence means that the series keeps growing larger without bound. For example, consider the geometric series:
\[ S = \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \text{...} \] This series converges to 1. However, the harmonic series we discussed does not converge. It diverges, even though the individual terms get smaller. This characteristic often surprises students. For the series given in the exercise, starting from \[ \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \text{...} \] each term contributes progressively smaller amounts, but the sum grows without bound. Determining whether a series converges or diverges is important in many fields, including analysis and number theory. There are various tests and criteria, such as the ratio test or the nth-term test, that help in analyzing series behavior.