Series - Convergence tests

1. The Comparison Test
Let $\displaystyle \sum_{n=1}^\infty a_n$ and $\displaystyle \sum_{n=1}^\infty b_n$ be series such that $0< a_n\le b_n$
* If $\displaystyle \sum_{n=1}^\infty b_n$ is Convergent then $\displaystyle \sum_{n=1}^\infty a_n$ is also convergent .
*  If $\displaystyle \sum_{n=1}^\infty a_n$ is divergent then $\displaystyle \sum_{n=1}^\infty b_n$  is also divergent .
2. The Limit comparison Test
Let $\displaystyle \sum_{n=1}^\infty a_n$ and $\displaystyle \sum_{n=1}^\infty b_n$ be series such that $a_n$ and $b_n$ are positive for all n .
*   If $0< \displaystyle \lim_{n \to \infty} {a_n \over b_n} < \infty$ then $\displaystyle \sum_{n=1}^\infty a_n$ and $\displaystyle \sum_{n=1}^\infty b_n$ are either both convergeent or both divergent
*  If $\displaystyle \lim_{n \to \infty} {a_n \over b_n} =0$ then $\displaystyle \sum_{n=1}^\infty b_n$ convergent implies that $\displaystyle \sum_{n=1}^\infty a_n$ is also convergent
*  If $\displaystyle \lim_{n \to \infty} {a_n \over b_n} =\infty$ then $\displaystyle \sum_{n=1}^\infty b_n$ divergent implies that $\displaystyle \sum_{n=1}^\infty a_n$ is also diverhent
3. p-series
p-series $\displaystyle \sum_{n=1}^\infty {1 \over n^p}$ converges for p > 1 and diverges for $0 < p\le 1.$
4. The Integral Test
Let $f(x)$ be a function which is continuous , positive , decreasing for all $x \ge 1.$ The series
$\displaystyle \sum_{n=1}^\infty f(n)= d(1)+f(2)+f(3)+...+(f(n)+...$
Converges if $\int \limits_1^ \infty f(x) dx$ converges, and diverges if
$\int \limits_1^ \infty f(x) dx \to \infty$ as $n \to \infty$
5. The Ratio Test
Let $\displaystyle \sum_{n=1}^\infty a_n$ be a series with positive terms .
If $\displaystyle \lim_{n \to \infty } {a_{n+1} \over a_n}< 1$ then $\displaystyle \sum_{n=1}^\infty a_n$ is convergent .
If $\displaystyle \lim_{n \to \infty } {a_{n+1} \over a_n}> 1$ then $\displaystyle \sum_{n=1}^\infty a_n$ is divergent.
If $\displaystyle \lim_{n \to \infty } {a_{n+1} \over a_n}= 1$ then $\displaystyle \sum_{n=1}^\infty a_n$ may converge or diverge and the ratio test is inconclusive ; some other tests must be used .  . .
6. The Root Test
Let $\displaystyle \sum_{n=1}^\infty a_n$ be a series with positive terms .
If $\displaystyle \lim_{n \to \infty } \sqrt [n]{a_n}<1$ then $\displaystyle \sum_{n=1}^\infty a_n$ is convergent .
If $\displaystyle \lim_{n \to \infty } \sqrt [n]{a_n}>1$ then $\displaystyle \sum_{n=1}^\infty a_n$ is divergent.
If $\displaystyle \lim_{n \to \infty } \sqrt [n]{a_n} = 1$ then $\displaystyle \sum_{n=1}^\infty a_n$ may converge or diverge , but no conclusion can be drawn from this test .