19.1 Sequences and Series

Section 19.1: 3, 4, 7, 9, 11, 14, 18, 19, 22, 23, 27, 29.

A sequence $\{z_n\}$ converges if $\lim\limits_{n\to\infty}z_n=L$, where $L$ is a complex number. In other words, $\text{Re}(z_n)$ converges to $\text{Re}(L)$, and $\text{Im}(z_n)$ converges to $\text{Im}(L)$.

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Geometric Series

Defined as:

$$ \sum_{k=1}^{\infty}az^{k-1}=a+az+az^2+az^3+... $$

When $|z| < 1$, the series converges to the following, and diverges for $|z| ≥1$

$$ S_\infty = \frac{a}{1-z} $$

Power Series

A power series centred at $z_0$ is of the form below, where $a_k$ coefficient is a complex constant.

$$ \sum_{k=1}^{\infty}a_k(z-z_0)^k $$

Circle of Convergence

Circle of convergence is a circle of radius $R$ centered at $z_0: |z-z_0| = R$

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$$ (i) \quad \lim_{{n \to \infty}} \left| \frac{{a_{n+1}}}{{a_n}} \right| = L \neq 0, \text{ the radius of convergence is } R = \frac{1}{L};

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$$ (ii)\quad \lim_{{n \to \infty}} \left| \frac{{a_{n+1}}}{{a_n}} \right| = 0, \text{ the radius of convergence is } R = \infty; $$

$$ (iii) \quad \lim_{{n \to \infty}} \left| \frac{{a_{n+1}}}{{a_n}} \right| = \infty, \text{ the radius of convergence is } R = 0. $$

19.2 Taylor Series