Final answer:
The series given in the form 1/n(n-1) is a telescoping series which simplifies to (n-1)/n. The solution relies on terms canceling each other out, resulting in a series that approaches 1 as n increases.
Step-by-step explanation:
The question asks to solve the series of terms in the form 1/n(n-1). This can be thought of as a telescoping series, where each fraction is comprised of a difference of two terms, such as 1/2 - 1/3 for the first term. By simplifying each fraction in such a way, we see that many middle terms cancel out, leaving us with only the first part of the first term and the second part of the last term. Therefore, the sum of such a series becomes 1 - 1/n, which simplifies to (n-1)/n or n/n - 1/n.
Using this approach, for any positive integer n, the sum of the series 1/2x1 + 1/2x3 + 1/4x3 ... + 1/n(n-1) simplifies to (n-1)/n, which represents a sequence approaching 1 as n becomes very large.
Understanding the properties of series expansions like this is crucial for mathematical analysis and can be a challenging but rewarding part of higher-level mathematics studies.