(Sum of a geometric series)

Let $x \in \mathbb{R}, n \in \mathbb{N}$. Depending on the value of $x$, determine the value of $\sum_{k=0}^{n} x^{k}$.

Proposition 1.1.13.

Let $a_{1}, a_{2}, \ldots, a_{n} \in \mathbb{R}$. Then $\left(\sum_{k=1}^{n} a_{k}\right)^{2}=\sum_{k=1}^{n} a_{k}^{2}+2 \sum_{1 \leq i<j \leq n} a_{i} a_{j}$

## - Manipulation of inequalities -

## Lemma 1.1.14.

For all $a, b, c$ and $d \in \mathbb{R}: a \leqslant c$ and $b \leqslant d \Rightarrow a+b \leqslant c+d$

## Lemma 1.1.15.

For all $a, b \in \mathbb{R}, k \in \mathbb{R}_{+}^{*}: a \leqslant b \Leftrightarrow k a \leqslant k b$

## Lemma 1.1.16.

For all $a, b \in \mathbb{R}, k \in \mathbb{R}_{-}^{*}: a \leqslant b \Leftrightarrow k a \geqslant k b$

## Lemma 1.1.17.

Let $f: A \rightarrow \mathbb{R}$ be an increasing function on an interval $B \subseteq A$.

Then for all $a, b \in B: a \leq b \Rightarrow f(a) \leq f(b)$.

If in addition $f$ is strictly increasing we have: $a \leq b \Leftrightarrow f(a) \leq f(b)$.

## Lemma 1.1.18.

Let $f: A \rightarrow \mathbb{R}$ be a decreasing function on an interval $B \subseteq A$.

Then for all $a, b \in B: a \leq b \Rightarrow f(a) \geq f(b)$.

If in addition $f$ is strictly decreasing we have: $a \leq b \Leftrightarrow f(a) \geq f(b)$.

## Remark 1.1.19.

The sum of positive terms is therefore always positive. Moreover, the sum is zero if and only if all the terms are zero.

## - A square is always positive -

## Proposition 1.1.20.

For all $x \in \mathbb{R}, x^{2} \geqslant 0$ with equality if and only if $x=0$

## Example 1.1.21.

Thus we have directly that for all $a, b \in \mathbb{R}: a^{2}+b^{2} \geqslant 2 a b$ with equality if and only if $a=b$.
For $x=1, \sum_{k=0}^{n} x^{k}=\sum_{k=0}^{n} 1=n+1$

For $x \neq 1$, use Remark 8: $\sum_{k=0}^{n} x^{k}=\frac{x^{n+1}-1}{x-1}$