**Sequence:** A list of values $a_i$ where each element is a function $a$ with input $i$, that returns $a_i$ as output. ^7e111e - where $i \in \mathbb N$ as the index $\{1, \dots , n\}$ - where $a_i \in \mathbb R$ as the values $ \text{sequence}=[a_1, \dots ,a_n] \implies a(i) = a_i $ **Series:** The summation of terms of a sequence. **Limits:** A sequence can converge to a certain value $a$ as soon as $i$ is sufficiently large. We say that $a_i \to a$, when $i\to \infty$. Expressed in limit notation it looks as follows.. $ \lim_{i \to \infty} a_i =a $ ![[converging-series.png]] ## Monotonic Series In a monotonic series $a_i < a_{i+1}$ for all $i$, we know that it either converges to some value $a$ or to infinity. $ \text{when } i \to \infty\begin{cases} a_i \to \infty \\a_i \to a \end{cases} $ To analyze which of the two cases is true, we can observe the distance to an assumed converging value for various $i$ values. The absolute difference is expressed as a series of its own $b_i$. When this series converges to $0$, then we know that $a_i$ converges to some finite value $a$. $ \text{if: } \lvert a_i -a \rvert \le b_i \quad \text{and} \quad b_i \to 0 \quad \text{then}\quad a_i\to a $ ## Sandwich Argument Let us assume that there are 2 series that both converge to some values $a,b$. If there is a third series $c_i$, whose values are always between $a_i, b_i$, the converging value $c$ will also be between $a,b$. ![[sandwich-argument.png]] ## Geometric Series A geometric series is the sum of an infinite number of terms that have a constant ratio between successive terms. This means the change from $a_i$ to $a_{i+1}$ always has the same factor of change. **Example:** In the below sequence the constant ratio is $\frac{1}{2}$, which is also denoted as $a$ in this regard. $ \begin{align} s &= 1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+ \cdots \\[6pt] s&=1+a+a^2+a^3+\cdots \text{ where } a=1/2 \end{align} $ **Convergence:** The series is the sum of this infinite sequence. For it to converge to some finite value, $\lvert a \rvert <1$. If this is the case, the limit of $S$ can be found with a simple formula. $ \begin{align} S&=\sum_{i=0}^\infty a_i= 1 + a + a^2 + \dots \\[6pt] S&= \frac{1}{1-a} \end{align} $