Absolute Convergence Test and Properties

Subjects: Real Analysis
Links: Series in R

If $\sum | a_{n} |$ converges, then $\sum a_{n}$ it is said that it converges absolutely. If $\sum | a_{n} |$ diverges but $\sum a_{n}$ still converges, then $\sum a_{n}$ is said to it converges conditionally.

Rearrangements

Definition:

Let $f : N \to N$ , where $f$ is bijective, $b_{n} = a_{f (n)}$ then, $\sum b_{n}$ is a rearrangements of $\sum a_{n}$

Theorem: If the series $\sum a_{n}$ is absolutely convergent, then any rearrangements of this series will converge to the same limit.

Riemann Rearrangement Theorem

Let $\sum_{n = 1}^{\infty} a_{n}$ an be a conditionally convergent series.

For every $x \in \overset{―}{R}$ there exists a rearrangement $σ$ such that:
$\sum_{n = 1}^{\infty} a_{σ (n)} = x$
There’s a rearrangement $σ$ such that $\sum_{n = 1}^{\infty} a_{σ (n)}$ doesn’t converge in $\overset{―}{R}$ .

Limit Comparison V.2

Let the sequences $(a_{n})$ , and $(b_{n})$ be non-zero, and that the following limit exists:

r = lim_{n \to \infty} | \frac{a_{n}}{b_{n}} |

If $r \neq 0$ , then $\sum a_{n}$ absolutely converges if and only if $\sum b_{n}$ absolutely converges.
If $r = 0$ , then if $\sum b_{n}$ absolutely converges, then, $\sum a_{n}$ absolutely converges.

Ratio and Root Test

Let the sequence $(a_{n})$ , given the following limits:

lim_{n \to \infty} | \frac{a_{n + 1}}{a_{n}} | = lim_{n \to \infty} | a_{n} |^{1 / n} = r

If $r < 1$ then the series $\sum a_{n}$ converges absolutely.
If $r > 1$ then the series $\sum a_{n}$ diverges.
If $r = 1$ , then the tests are inconclusive

Raabe’s Test

If the Ratio and Root Test are inclusive, then by checking the following limit, it can help:

r^{'} = lim_{n \to \infty} (n (1 - | \frac{a_{n + 1}}{a_{n}} |))

If $r^{'} < 1$ then the series $\sum a_{n}$ converges absolutely.
If $r^{'} > 1$ then the series $\sum a_{n}$ diverges.
If $r^{'} = 1$ , then the test is inconclusive.