# Using the integral test, how do you show whether #sum1/(n^2 - 1)# diverges or converges from n=4 to infinity?

by using the Quotient rule we get:

now we know we can use the integral test for convergence.

where you can use Partial Fractions to get:

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By using the integral test, we can determine whether the series ∑(1/(n^2 - 1)) from n = 4 to infinity converges or diverges.

First, we rewrite the series as ∑(1/(n + 1)(n - 1)).

Next, we consider the function f(x) = 1/(x + 1)(x - 1).

The function f(x) is continuous, positive, and decreasing for x ≥ 4.

We can then integrate f(x) from 4 to infinity to find the integral ∫(1/(x + 1)(x - 1)) dx.

This integral can be evaluated using partial fractions, which gives us the integral ∫(1/(2(x - 1)) - 1/(2(x + 1))) dx.

Evaluating this integral from 4 to infinity gives us lim as b approaches infinity of (ln|2(b - 1)| - ln|2(4 - 1)|), which simplifies to lim as b approaches infinity of (ln|b - 1| - ln(3)).

Since the limit of ln|b - 1| as b approaches infinity is infinity, the integral diverges.

Therefore, by the integral test, the series ∑(1/(n^2 - 1)) from n = 4 to infinity also diverges.

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