series

consider the following series $\text{[math]}$

consider the series:
$\text{[math]}$ this series converges to 1 as it ends with $\text{[math]}$

$\text{[math]}$ this series diverges

a necessary condition for the convergence of a series is $\text{[math]}$

consider $\text{[math]}$ , the first diverges but the second converges therefore $\text{[math]}$

the sum $\text{[math]}$ converges when $\text{[math]}$ and diverges when $\text{[math]}$