Sequences and Series

Sequences

A sequence is a string of indexed numbers $a_1, a_2, a_3, \ldots$. We denote this sequence with $(a_n)_{n\geq1}$.

Details

In a sequence the same number can appear several times in different places.

Examples

Example

$(\displaystyle\frac{1}{n})_{n\geq1} \text{ is the sequence } 1, \displaystyle\frac{1}{2}, \displaystyle\frac{1}{3}, \displaystyle\frac{1}{4}, \ldots$

Example

$(n)_{n\geq1} \text{ is the sequence } 1, 2, 3, 4, 5, \ldots$

Example

$(2^nn)_{n\geq1} \text{ is the sequence } 2, 8, 24, 64, \ldots$

Convergent Sequences

A sequence $a_n$ is said to converge to the number $b$ if for every $\varepsilon >0$ we can find an $N\in \mathbb{N}$ such that $|a_n-b| < \varepsilon$ for all $n \geq N$. We denote this with $\lim_{n\to\infty}a_n=b$ or $a_n\to b$, as $n\to\infty$.

Details

A sequence $a_n$ is said to converge to the number $b$ if for every $\varepsilon >0$ we can find an $N\in \mathbb{N}$ such that $|a_n-b| < \varepsilon$ for all $n \geq N$. We denote this with $\lim_{n\to\infty}a_n=b$ or $a_n\to b$, as $n\to\infty$.

If $x$ is a number, then

$(1 + \displaystyle\frac{x}{n})^n \rightarrow e^x \text{ as } n\rightarrow\infty$

Examples

Example

The sequence $(\displaystyle\frac{1}{n})_{n\geq\infty}$ converges to $0$ as $n\to\infty$.

Example

If x is a number, then

$(1 + \displaystyle\frac{x}{n})^n \rightarrow e^x \text{ as } n\rightarrow\infty$

Infinite Sums (series)

We are interested in, whether infinite sums of sequences can be defined.

Details

Consider a sequence of numbers, $(a_n)_{n\to\infty}$.

Now define another sequence $(s_n)_{n\to\infty},$ where

$s_n=\displaystyle\sum_{k=1}^na_k$

If $(s_n)_{n\to\infty}$ is convergent to $S=\lim_{n\to\infty}s_n,$ then we write

$S=\displaystyle\sum_{n=1}^{\infty}a_n$

Examples

Example

If

$a_k = x^k, qquad k=0,1,\dots$

then

$s_n=\displaystyle\sum_{k=0}^{n}x^k=x^0+x^1+\dots.+x^n$

Note also that

$xs_n=x(x^0+x^1+\dots.+x^n)= x + x^2 + \dots + x^{n+1}$

We have

$s_n = 1 + x + x^2 + \dots + x^n$

$xs_n = x + x^2 + \dots +x^n + x^{n+1}$

$s_n – xs_n = 1 - x^{n+1}$

i.e.

$s_n(1-x) = 1-x^{n+1}$

and we have

$s_n =\displaystyle\frac{1-x^{n+1}}{1-x}$

if $x\neq1$.

If $0< x<1$ then $x^{n+1}\to 0$ as $n\to\infty$ and we obtain $s_n\to\displaystyle\frac{1}{1-x}$ so

$\displaystyle\sum_{n=0}^{\infty}x^n=\displaystyle\frac{1}{1-x}$

The Exponential Function and the Poisson Distribution

The exponential function can be written as a series (infinite sum):

$e^x=\displaystyle\sum_{n=0}^{\infty}\displaystyle\frac{x^n}{n!}$

The Poisson distribution is defined by the probabilities

$p(x)=e^{-\lambda}\displaystyle\frac{\lambda^x}{x!}\textrm{ for } x=0,\ 1,\ 2,\ \ldots$

Details

The exponential function can be written as a series (infinite sum):

$e^x=\displaystyle\sum_{n=0}^{\infty}\displaystyle\frac{x^n}{n!}$

Knowing this we can see why the Poisson probabilities

$p(x)=e^{-\lambda}\displaystyle\frac{\lambda^x}{x!}$

add to one:

$\displaystyle\sum_{x=0}^{\infty}p(x)=\displaystyle\sum_{x=0}^{\infty}e^{-\lambda}\displaystyle\frac{\lambda^x}{x!}=e^{-\lambda}\displaystyle\sum_{x=0}^{\infty}\displaystyle\frac{\lambda^x}{x!}=e^{-\lambda}e^{\lambda}=1$

Relation to Expected Values

The expected value for the Poisson is given by

$$\begin{aligned} \displaystyle\sum_{x=0}^\infty x p(x) &= \displaystyle\sum_{x=0}^\infty x e^{-\lambda} \displaystyle\frac{\lambda^x}{x!} \\ &= \lambda \end{aligned}$$

Details

The expected value for the Poisson is given by

$$\begin{aligned} \displaystyle\sum_{x=0}^\infty x p(x) &= \displaystyle\sum_{x=0}^\infty x e^{-\lambda} \displaystyle\frac{\lambda^x}{x!} \\ &= e^{-\lambda} \displaystyle\sum_{x=1}^\infty \displaystyle\frac{x\lambda^x}{x!} \\ &= e^{-\lambda} \displaystyle\sum_{x=1}^\infty \displaystyle\frac{\lambda^x}{(x-1)!} \\ &= e^{-\lambda} \lambda \displaystyle\sum_{x=1}^\infty \displaystyle\frac{\lambda^{(x-1)}}{(x-1)!} \\ &= e^{-\lambda} \lambda \displaystyle\sum_{x=0}^\infty \displaystyle\frac{\lambda^{x}}{x!} \\ &= e^{-\lambda} \lambda e^{\lambda} \\ &= \lambda \end{aligned}$$