I have checked many of this site's pages yet I could not find a clear answer about how to choose between the "comparison test" OR the "limit comparison test".
Because the difference between the two is that in the limit comparison test, one has to evaluate the limit as n goes to infinity of $a_n/b_n$. Where in the comparison test or 'direct comparison test' would that NOT be necessary?
Thanks in advance.
Say we have two series, $\sum a_n$ and $\sum b_n$.
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The comparison test requires $a_n \le b_n$ (or vice-versa) for all $n$ and $a_n, b_n \ge 0$ for all $n$.
Then, if $\sum b_n$ is convergent, so is $\sum a_n$. If $\sum a_n$ is divergent, then so it $\sum b_n$.
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The limit comparison test does not require that $a_n \le b_n$ (or vice-versa) for all $n$.
So, if you have $a_n \ge 0, b_n > 0$ for all $n$, you define $c = \lim_{n \to \infty}\frac{a_n}{b_n}$.
If $c$ is positive and finite then either both diverge or both converge.
Explanation: The comparison test is easier to implement, but it has a strict requirement that the limit comparison test does not have ($a_n \le b_n$ or vice-versa for all $n$). This is useful because it allows one series to serve as a bound for the other.
If this condition is not met, then you should use the limit comparison test, as it does not have this requirement.
