Series

A series is an infinite sum of values in a sequence. Series are usually written in summation notation, using capital Sigma ($\Sigma$). Series are useful to calculate approximations with high accuracy to irrational values (such as Pi) as well as other fields such as digital signal processing in computer science.

Definition §

A series is obtained by adding the values of a sequence. Remember that sequence notation is represented as $a_n$ Written in summation notation:

$$a_1+a_2+a_3+\cdots =\sum _{n=1}^{\infty }{a}_n$$

Partial Sums §

A partial sum of a series is simply a finite sum subset of the series. I.e. by adding the first $n$ values of a series, we obtain the $n$th partial sum.

$$s_k=\sum _{n=1}^k{a}_n$$

Convergence Of A Series §

Remember how series are infinite sums? Well the same concept of limits can be applied onto series.

Let $a_n$ be a sequence, with its corresponding sequence of partial sums being $s_n$.

The sequence of partial sums as n approaches infinity (i.e. a series) may or may not have a limit. If it does, however, then the limit of the partial sums can be used to find the convergence of the sum of the series:

$$\sum _{n=1}^{\infty }{a}_n=\underset{n\to \infty }{\lim }{s}_n=S$$

If the sequence of partial sums does not have a limit, then the series is said to diverge.

Properties Of Series §

Let ${\sum }_{n=1}^{\infty }{a}_n$ and ${\sum }_{n=1}^{\infty }{b}_n$ be two unique series that both converge onto values $S_a$ and $S_b$ respectively and let $c\in \mathbb{R}\backslash \{0\}$ be a constant:

1. ${\sum }_{n=1}^{\infty }(a_n+b_n)$ = ${\sum }_{n=1}^{\infty }{a}_n+{\sum }_{n=1}^{\infty }{b}_n=S_a+S_b$ [converges]

2. ${\sum }_{n=1}^{\infty }c\times a_n$ = $c\times {\sum }_{n=1}^{\infty }{a}_n$ = $c\times S_a$ [converges]

3. If ${\sum }_{n=1}^{\infty }{f}_n$ is a series that diverges, then ${\sum }_{n=1}^{\infty }c\times f_n$ also diverges ($c\ne 0$)

Convergence/Divergence Tests §

Conditions	Test
The limit of $a_n$ seems to equate to 0	Divergence Test
Series contains factorials and/or exponents	Ratio Test
Else	Comparison Test

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• Comparison Test
• Divergence Test
• Ratio Test
• Geometric Series
• P-Harmonic Series

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