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:

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

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

Geometric Series §

A geometric series is a series in which future terms are generated by the multiplication of a common factor, $r$. It is always in the form:

${\sum }_{n=1}^{\infty }a\times r^{n-1}=ar+a{r}^2+a{r}^3+\dots a{r}^{n-1}$

where $a\in \mathbb{R}\backslash \{0\}$ and $r\in \mathbb{R}$

The series converges if $|r|<1$ and diverges if $|r|\ge 1$. Since we are repeatedly multiplying by a number smaller than 1, we are essentially reducing the value of future terms, and hence the final sum converges into a value. If $|r|<1$ then:

${\sum }_{n=0}^{\infty }a{r}^n=\frac{a}{1-r}$ This formula is obtained by the general formula for geometric series: ${\sum }_{k=0}^{n}a{r}^k=\frac{a(1-r^{n+1})}{1-r}$

p-Harmonic Series §

A p-harmonic series is a series consisting of unit fractions (i.e. fractions smaller than 1). It is of the form:

${\sum }_{n=1}^{\infty }\frac{1}{n^p}$ If $p>1$, then the series converges to 0. Else, if $p\le 1$, the series diverges.

Properties Of Series (Convergence) §

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]

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