Convergence of Series

Checking convergence of a series comes down to

finding a general expression for $S_{n}$
determining if $lim_{n \to \infty} S_{n}$ exists and is finite

For most series, finding a general expression for $S_{n}$ is not possible, so we use different methods to determine if the series diverges or converges:

using tests
manipulating the series to get a basic series

Letting $\sum a_{n}$ and $\sum b_{n}$ be series wit positive terms:

Divergence Test

Or $n^{t h}$ Term Test

$lim_{n \to \infty} a_{n} \neq 0$ or $lim_{n \to \infty} a_{n}$ does not exist $\to$ $\sum_{n = 1}^{\infty} a_{n}$ diverges

Integral Test

Defining the general term of the series $\sum a_{n}$ as a function $f (x)$ where $a_{n} = f (n)$

If $f$ is continuous, positive and decreasing on the interval from the series $\to$ $\sum a_{n}$ is convergent $⟺$ $\int_{1}^{\infty} f (x) d x$ is convergent, i.e.

$\int_{1}^{\infty} f (x) d x$ converges $⟺$ $\sum_{1}^{\infty} a_{n}$ converges
$\int_{1}^{\infty} f (x) d x$ diverges $⟺ \sum_{n = 1}^{\infty} a_{n}$ diverges

Comparison Tests

(Direct) Comparison Tests exploit the monotone increasing/decreasing sequence theorems

0 < a_{n} \leq b_{n}, \forall n \geq 1 \to if \sum b_{n} converges then \sum a_{n} converges

0 < b_{n} \leq a_{n}, \forall n \geq 1 \to if \sum b_{n} diverges then \sum a_{n} diverges

Limit Comparison Test

If lim_{n \to \infty} \frac{a_{n}}{b_{n}} = c, 0 < c < \infty \to both series converge or diverge

if $c = 0$ and $\sum b_{n}$ converges, $\to \sum a_{n}$ converges
if $c = \infty$ and $\sum b_{n}$ diverges, $\to \sum a_{n}$ diverges

Alternating Series (Leibniz) Test

Series with terms that are positive and/or negative

like alternating between positive and negative

{a_{n}} is a positive, decreasing series with lim_{n \to \infty} a_{n} = 0 \to \sum_{n = 1}^{\infty} (- 1)^{n - 1} a_{n} converges

If $\sum | a_{n} |$ converges, then $\sum a_{n}$ converges

Absolutely Convergent Series

A series $\sum a_{n}$ where $\sum | a_{n} |$ converges

$\sum \frac{(- 1)^{n}}{n^{2}}$

Conditionally Convergent Series

A series where $\sum a_{n}$ converges, but $\sum | a_{n} |$ diverges

$\sum \frac{(- 1)^{n}}{n}$ converges but $\sum \frac{1}{n}$ diverges (p=1)

Ratio Test

With $\sum a_{n}$ and $L = lim_{n \to \infty} | \frac{a_{n + 1}}{a_{n}} |$

$L < 1 \to \sum a_{n}$ converges absolutely
$L > 1 \to \sum a_{n}$ diverges
$L = 1 \to$ Ratio Test is inconclusive

Root Test

With $\sum a_{n}$ and $L = lim_{n \to \infty} \sqrt[^{n}]{| a_{n} |}$

$L < 1 \to \sum a_{n}$ converges absolutely
$L > 1 \to \sum a_{n}$ diverges
$L = 1 \to$ Root Test is inconclusive

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