Add content for functions/sequences-series

Add actual content to the skeleton of the `functions/sequences-series`
section.

Signed-off-by: Eggert Karl Hafsteinsson <[email protected]>
Signed-off-by: Teodor Dutu <[email protected]>
Signed-off-by: Razvan Deaconescu <[email protected]>

chapters/functions/sequences-series/reading/README.md

@@ -1 +1,167 @@
 # 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
+
+:::info Example
+
+$$(\displaystyle\frac{1}{n})_{n\geq1} \text{is the sequence} 1, \displaystyle\frac{1}{2}, \displaystyle\frac{1}{3}, \displaystyle\frac{1}{4}, \ldots$$
+
+:::
+
+:::info Example
+
+$$(n)_{n\geq1} \text{is the sequence } 1, 2, 3, 4, 5, \ldots$$
+
+:::
+
+:::info 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
+
+:::info Example
+
+The sequence $(\displaystyle\frac{1}{n})_{n\geq\infty}$ converges to $0$ as $n\to\infty$.
+
+:::
+
+:::info 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
+
+:::info 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
+
+$$
+\begin{aligned}
+  \displaystyle\sum_{x=0}^\infty x p(x) &= \displaystyle\sum_{x=0}^\infty x e^{-\lambda} \displaystyle\frac{\lambda^x}{x!} \\
+  &= e^{-\lambda} \displaystyle\sum_{x=1}^\infty \displaystyle\frac{x\lambda^x}{x!} \\
+  &= e^{-\lambda} \displaystyle\sum_{x=1}^\infty \displaystyle\frac{\lambda^x}{(x-1)!} \\
+  &= e^{-\lambda} \lambda \displaystyle\sum_{x=1}^\infty \displaystyle\frac{\lambda^{(x-1)}}{(x-1)!} \\
+  &= e^{-\lambda} \lambda \displaystyle\sum_{x=0}^\infty \displaystyle\frac{\lambda^{x}}{x!} \\
+  &= e^{-\lambda} \lambda e^{\lambda} \\
+  &= \lambda
+\end{aligned}
+$$