For each of the following series, using no tests besides the nth Term and Comparison Tests, determine whether the series converges, diverges to pminfty, or diverges, not to pminfty sumfrac{n-1}{n^2-1}

Question

For each of the following series, using no tests besides the nth Term and Comparison Tests, determine whether the series converges, diverges to

\pm \infty

, or diverges, not to

\pm \infty

\sum \frac{n - 1}{n^{2} - 1}

Answer 1

The objective here is to determine whether the given series is convergent, diverges to $\pm \infty$ , or diverges, not to $\pm \infty$
Series is:
$\sum \frac{n - 1}{n^{2} - 1}$
First simplify the nth term of the series as:
$a_{n} = \frac{n - 1}{n^{2} - 1} = \frac{n - 1}{(n - 1) (n + 1)} = \frac{1}{n + 1}$
$\sum \frac{n - 1}{n^{2} - 1} = \sum \frac{1}{n + 1}$ , and
$\frac{1}{n} < \frac{1}{n + 1}$ and
$\sum \frac{1}{n}$ is convergent by p-test series
Thus, by comparison test series
$\sum \frac{1}{n + 1} = \sum \frac{n - 1}{n^{2} - 1}$ is convergent.
And it diverges to $\infty$ since the series $\sum \frac{1}{n}$ diverges to $\infty$
Thus, the given series diverges to $\infty$