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 and computer science.

All sequence henceforth are defined over the real numbers, i.e. $a_n:\mathbb{N}\to \mathbb{R}$

Definition §

Series

Let $(a_n{)}_{n\in \mathbb{N}}$ be a sequence. Then, a series has two definitions. It is defined as a sequence of partial sums:

$$\sum _{n=1}^{\infty }(a_n)=(s_n{)}_{n\in \mathbb{N}}$$

But, if the sequence of partial sums converges, then it is the converging point (denoted $S$):

$$\sum _{n=1}^{\infty }(a_n)=\underset{n\to \infty }{\lim }(s_n)=S$$

If such a limit does not exist, the series diverges. Else it converges to $S$

Partial Sums §

Partial Sum

Let $(a_n{)}_{n\in \mathbb{N}}$ be a sequence. Then, a (sequence of) partial sums $s_k$ is defined as:

$$(s_n{)}_{n\in \mathbb{N}},s_n=\sum _{i=1}^n{a}_i=a_1+a_2+\cdots +a_n$$

Proofs & Lemmas §

L1: If the sequence of partial sums for a sequence with non-negative terms is bounded, it is convergent ^L1

Let $(a_n{)}_{n\in \mathbb{N}}$ be a sequence with non-negative terms, that is, $a_n\ge 0\forall n\in \mathbb{N}$. Let it’s corresponding sequence of partial sums be $(s_k{)}_{k\in \mathbb{N}}$. Then:

$$(s_k{)}_{k\in \mathbb{N}}\text{ is a bounded sequence}\to (a_n{)}_{n\in \mathbb{N}}\text{ is convergent}$$

Alternatively, a series with positive terms converges iff it is bounded

Proof $s_k$ as:

We can write any term

$$s_k=s_{k-1}+a_n$$

Since $a_n\ge 0$, by the ordered field axioms, we know:

$$s_{k-1}+a_n\ge s_{k-1}\to s_k\ge s_{k-1}\forall k\in \mathbb{N}$$

This is the definition of a monotone increasing sequence.

Since we assume that $(s_k{)}_{k\in \mathbb{N}}$ is a bounded sequence, we can use the lemma that all bounded monotonic sequences converge, namely,

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

Examples §

1. $a_n=1$, then $s_n=n$

${\sum }_{n=1}^{\infty }{a}_n$ diverges, as $s_n$ is unbounded.

2. $a_n=n$, then $s_n=\frac{n(n+1)}{2}$ , see triangular numbers

${\sum }_{n=1}^{\infty }{a}_n$ diverges for the same reason as (1.)

3. $a_n=\frac{1}{n!}$

${\sum }_{n=1}^{\infty }{a}_n$ converges to $e$ (Euler’s number)

The sequence is non-negative (since $0!=1>0$ and $(k+1)!\ge k!$)

Using a lemma that $n!\ge 2^n$, which implies $\frac{1}{n!}<\frac{1}{2^n}$

Then the partial sum:

$$s_n=\sum _{k=0}^n(\frac{1}{k!})=1+1+\frac{1}{2}+\frac{1}{6}+\dots$$

Exactly, it is:

$$\begin{array}{rl}{s}_n& =1+1+\frac{1}{2}+\frac{1}{6}+\sum _{k=4}^n(\frac{1}{k!})\\ & =\frac{8}{3}+\sum _{k=4}^n(\frac{1}{k!})\\ & \le \frac{8}{3}+\sum _{k=4}^{\infty }(\frac{1}{2^k})\\ & =\frac{8}{3}+\left(\sum _{k=1}^{\infty }(\frac{1}{2^k})-\sum _{k=1}^4(\frac{1}{2^k})\right)\\ & \le (\frac{8}{3})+(1-\frac{7}{8})\\ & =\frac{8}{3}+\frac{1}{8}=2\frac{19}{24}\end{array}\text{\hspace{1em}(2k1 is a geometric series →1)}$$

Which shows it has a upper bound of $2.7917$. Compared it to $e$, which is $2.71828$.

Since this series has a non-negative sequence and is bounded, it is convergent by L1

Example: Geometric Series §

Example: Harmonic & P-Harmonic Series §

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

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