# 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}

For each of the following series, using no tests besides the nth Term and Comparison Tests, determine whether the series converges, diverges to $±\mathrm{\infty }$, or diverges, not to $±\mathrm{\infty }$
$\sum \frac{n-1}{{n}^{2}-1}$
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coffentw

The objective here is to determine whether the given series is convergent, diverges to $±\mathrm{\infty }$, or diverges, not to $±\mathrm{\infty }$
Series is:
$\sum \frac{n-1}{{n}^{2}-1}$
First simplify the nth term of the series as:
${a}_{n}=\frac{n-1}{{n}^{2}-1}=\frac{n-1}{\left(n-1\right)\left(n+1\right)}=\frac{1}{n+1}$
$\sum \frac{n-1}{{n}^{2}-1}=\sum \frac{1}{n+1}$, 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 $\mathrm{\infty }$ since the series $\sum \frac{1}{n}$ diverges to $\mathrm{\infty }$
Thus, the given series diverges to $\mathrm{\infty }$

Jeffrey Jordon