Final answer:
To solve the provided series, a pattern recognition approach is utilized where pairs of terms are imaginatively adjusted to reveal that the sum of the series simplifies to 2n². This method skips complex fractions combination and utilizes basic algebraic manipulation to identify that the sum is n squared.
Step-by-step explanation:
The problem provided by the student is a series in the form of fractions, where the numerators are constant (1) and the denominators are the product of sequential numbers (2, 3, 4,..., n). To find the sum of this series, a pattern must be recognized. By manipulating the terms of the series imaginatively such that you pair off terms (taking a part from the last and adding to the first, then taking from the penultimate to the second, and so on), it becomes clear that the sum of this series can be simplified to 2n².
Mathematically, this can be seen when you add (n − 1) to the first term of the series and proceed similarly with the other terms. The sum will consist of n pairs of n, thus resulting in 2n². This method uses the intuitive understanding of how numbers combine and how series work, leaning on the student's familiarity with basic algebraic manipulation and series expansion concepts, along with visualizing numerical patterns.
To solve this series, no common denominator or multiplication technique is needed as suggested in series expansions or fractions combination. Instead, the pattern recognition simplifies the process, confirming that the sum of the series is simply n squared (n²), under the assumption that the original expression aligns with the pattern provided.