Lecture 5

Triangle inequality

Triangle inequality (I): $\forall a,b \in \mathbb{R}$, $|a+b| \leq |a| + |b|$

Triangle inequality (II):$\forall a,b \in \mathbb{R}, |a - b| \leq |a| + |b|$

Proof of (II):

     Let $a,b \in \mathbb{R}$, $|a - b| = |a + (-b)| \leq |a| + |-b|= |a| + |b| \qquad \qquad \qquad \square$

Note: In general, "|$a-b$|" describes the "distance" from $a$ to $b$.

Corollary: $\forall a,b,c \in \mathbb{R}$, $|a-b| \leq |a-c| + |b-c|$

Proof:

Let $a'=a-c$ and $b' = b-c$. Then the triangle inequality says:
$|a'-b'| \leq |a'| + |b'|$
$|a -c - (b - c)| \leq |a -c| + |b - c|$
$|a-b| \leq |a-c| + |b-c| \qquad \qquad \qquad \square$

Propisition: let $a \in \mathbb{R}, r > 0$, then for every $x \in \mathbb{R}$:

$|x-a| < r \iff -r < x-a< r \iff a - r < x < a + r \iff x \in (a - r, a + r)$

Summation identities:

Difference of Powers: let $a,b \in \mathbb{R}$

for $n \in \mathbb{N}$: $a^n - b^n = (a -b)(a^{n-1} + a^{n-2} \cdot b + ... + a \cdot b^{n-2} + b^{n-1})$

$a^n - b^n = (a^{n} + a^{n-1} \cdot b + ... + a^2 \cdot b^{n-2} + a \cdot b^{n-1}) - (a^{n-1} \cdot b - a^{n-2} \cdot b^2 + ... - a \cdot b^{n-1} + b^{n})$

$a^n - b^n = a^n - b^n$

Geometric Sum:

For $x \in \mathbb{R}, n \in \mathbb{N}$

$1 + x + x^2 + ... + x^n = \frac{1 - x^{n+1}}{1-x}$

Proof:

$a,b \in \mathbb{R}, n \in \mathbb{N}.$

Pick $a = 1, b =x$:

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

$\implies 1 + x + ...x^n = (1 - x^{n+1}) / (1 - x)$

Althernate proof:

Let $x \neq 1, n \in \mathbb{N}$. Define $S_n = 1 + x + ... + x^n$

$x \cdot S_n = x + x^2 + ... +x^n + x^{n+1}$

$S_n - x \cdot S_n = 1 - x^{n+1}$

$(1 - x) \cdot S_n = 1 - x^{n+1}$

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

Binomial formula:

Define (factorial): $0! = 1$ by convention

for $n \in \mathbb{N}$, $n! = n \cdot (n - 1)! = n \cdot (n - 1) \cdot (n - 2) ... \cdot 2 \cdot 1$

Binomial coefficient: For $n \in \mathbb{N}, 0 \leq k \leq n$S:

$\binom{n}{k} = \frac{n!}{k!(n-k)!}$

Notice: $\binom{n}{n-k} = \frac{n!}{(n-k)!(n-(n-k))!} = \binom{n}{k}$

Binomial formula: $\forall a,b \in \mathbb{R}, n \in \mathbb{N}:$

$(a+b)^n = \binom{n}{0} a^{n} + \binom{n}{1} a^{n-1} b + ... + \binom{n}{n-1} ab^{n-1} + \binom{n}{n} b^{n}$

Writing sums: Let $c_1, c_2, ..., c_n$, for $n \in \mathbb{N}$.

We write the following: $c_1 + c_2 + ... + c_n = \sum_{k=1}^{n} c_k$. The summation of one to $n$ of $c_k$. $k$ is a dummy index.

Finite sums are linear:

If $b_1,...,b_n, c_1,...,c_n \in \mathbb{R}$, then $\sum_{k=1}^{n} (c_k + b_k) = \sum_{k=1}^{n} c_k + \sum_{k=1}^{n} b_k$

If $M \in \mathbb{R}, \sum_{k=1}^{n} M \cdot b_k = M \sum_{k=1}^{n} b_k$

Rewrite the three previous identities using summations:

Difference of powers: $a,b \in \mathbb{R}, n \in \mathbb{N}, a^n - b^n = (a-b) \sum_{k=0}^{n-1} {a^{n-1-k} b^{k}}$

Geometric sum: $\forall x \in \mathbb{R}, \forall n \in \mathbb{N}, \sum_{k=0}^{n} {x^{k}} = \frac{1-x^{n+1}}{1-x}$

Binomial formula: $\forall a,b \in \mathbb{R}, n \in \mathbb{N}, (a+b)^n = \sum_{k=0}^{n} {\binom{n}{k}a^{n-k}b^{k}}$

Sequences

Definition: A sequence of real numbers is a real valued function $f$ such that, $f: \mathbb{N} \rightarrow \mathbb{R} n \rightarrow f(n)$

All of these mean the same thing:

• ${(a_{n})}_{n \in \mathbb{N}}$
• ${(a_{n})}$
• ${(a_{n})}_{n}$

Example: $a_n = \frac{1}{n}$, n \in \mathbb{N}$. $a_1 = 1, a_2 = \frac{1}{2}, a_3 = \frac{1}{3}, ...$

Example: $a_n = (-1)^{n}, n \in \mathbb{N}$. $a_1 = -1, a_2 = 1, a_3 = -1, ...$