Final answer:
The question asks for determining the convergence or divergence of a series by expressing it as a telescoping sum. However, the expression given, 4n^2 - 1, cannot be easily decomposed into such a form to reveal a telescoping structure, hinting at divergence. Other tests for convergence would be required to determine the behavior of the series.
Step-by-step explanation:
To determine whether the series ∑4 n2 - 1 from n = 3 to infinity is convergent or divergent by expressing sn as a telescoping sum, we need to find a way to decompose the general term into partial fractions that will cancel each other out as n progresses. This involves finding numbers a and b such that:
4n2 - 1 = (a/n + b) - (a/n + b + 1)
By refining this decomposition, we notice that when summed over n, most terms will cancel, leaving only a few from the beginning and end of the series. If the few terms that remain consist of constants or terms that go to zero as n approaches infinity, the series is convergent; otherwise, it is divergent. With our n2 term, the series appears to be a sum of positive integers, increasing without bound, hinting at divergence.
In practice, we look at the series and attempt to write it in the form:
4n2 - 1 = (2n - 1)(2n + 1)
We would then explore a telescoping property, where terms in sequence successively cancel each other, such as in:
1/(n(n + 1)) = 1/n - 1/(n + 1)
However, with a sequence like 4n2 - 1, we can't immediately find a telescoping form that simplifies in such a manner. As a result, we can't express the sum as a telescoping series using simple factors, and we'd need to use other convergence tests to determine the behavior of the series.