Definition:
$$ \begin{aligned} u_1, u_2, u_3, \ldots, u_N &\quad u_0, u_1, u_2, \ldots, u_{N-1} \\ u_{n+1}=f(u_n)&\quad u_1=a \end{aligned} $$
🪢 Sum series:
$$ \begin{aligned} S_N&=u_1+u_2+u_3+\ldots+u_N\\ &=\sum_{n=1}^{N}u_i \end{aligned} $$
🦷 Product series:
$$ \prod_{n=1}^Nu_n=u_1u_2u_3\ldots u_N $$
Cases:
$$ \begin{aligned} u_{n+1}=u_N+r&\Rightarrow u_n=u_a+(n-a)r \\ \sum^N_{n=b}u_a+(n-a)r= S_N&=\frac{1}{2}(N-b+1)(u_b+u_N) \end{aligned} $$
$$ \begin{aligned} v_{n+1}=qv_n&\Rightarrow v_n=v_aq^{n-a} \\ \sum_{n=b}^N v_aq^{n-a}=S_N&=\frac{v_b\left(1-q^{N+1-b}\right)}{1-q} \\ \lim_{\begin{aligned}\small{N\to\infin} \\ \small{q\ge1} \end{aligned}}S_N = \pm\infin &\quad \lim_{\begin{aligned}\small{N\to\infin} \\ \small{|q|\le1} \end{aligned}}S_N = \frac{v_0}{1-q} \end{aligned} $$
$$ \begin{aligned} (a+b)^n&=a^n+na^{n-1}b+\frac{n(n-1)}{2!}aN^{n-2}b^2+\ldots+b^n\\ &=\sum_{r=0}^{n}\frac{n!}{r!(n-r)!}a^{n-r}b^r \\ &= \sum_{r=0}^{n} \binom{n}{r}a^{n-r}b^r \end{aligned} $$
Case: we can use this to assume an approximation of the value of a number when $(1+x)^n$ when $|x|<1$:
$$ \left.(1+x)^n\right|_{|x|\ll 1}\approx 1 + nx $$
<aside> <img src="https://s3-us-west-2.amazonaws.com/secure.notion-static.com/22820bb7-7236-4a27-adc6-1ded5eb837e4/Taylor_series.png" alt="https://s3-us-west-2.amazonaws.com/secure.notion-static.com/22820bb7-7236-4a27-adc6-1ded5eb837e4/Taylor_series.png" width="40px" /> For a function that meets the following conditions: • single valued • continuous • $N$-times differentiable a function can be aproximated as:
$$ \begin{aligned} f(x)&\approx f(a)+\frac{f^{\prime}}{1!}(x-a)+\frac{f^{\prime\prime}(a)}{2!}(x-a)^2+\ldots+\frac{f^{(N)}(a)}{N!}(x-a)^N \\ &=\sum_{n=0}^{\infty}\frac{f^{(n)}(a)}{n!}(x-a)^n \end{aligned} $$
</aside>
https://www.desmos.com/calculator/zvvhu3bouz
Case: The Maclaurin series is the special case of the Taylot series when ****$a=0$
