Series part 2

Question 1

Taylor’s theorem

Answer 1

Let $z_0$ ∈ C and $R_0 > 0$, and suppose that
a complex function f is analytic on the disk
$D(z_0, R_0)$. Then
$f(z) = \sum_{n=0}^{\infty} a_n(z−z_0)^n$ whenever $|z−z_0| < R_0$,

where $a_n = \frac{f^{(n)}(z_0)}{n!}
(n = 0, 1, 2, . . .).$
This series is called the Taylor series of f at
$z_0$.

Question 2

When does the taylor series of a function converge

Answer 2

the Taylor series of f converges to f at each point within the circle with centre z0 whose radius is the distance from z0 to the nearest point z1 at which f is not analytic.

Question 3

Maclaurin series for $\frac{1}{1-z}$

Answer 3

$\frac{1}{1-z} = \sum_{n=0}^{\infty} z^n = 1 + z + z^2 +z^3 + …$
for $|z|<1$

Question 4

Maclaurin series for $e^z$

Answer 4

$e^z$ is entire so it = its Maclauirn (and Taylor) seireis on C

$\sum_{n=0}^{\infty} e^z = \frac{z^n}{n!} = 1 + z + \frac{z^2}{2} + \frac{z^3}{3!} + … $
$\forall z \in \ C$

Question 5

Maclaurin series for $sinz$

Answer 5

$ \sin{z} = \sum_{n=0}^{\infty} blahh = z - \frac{z^3}{3!} + \frac{z^5}{5!} - \frac{z^7}{7!} + …$

$\forall \ z \in \ C$ since sinz is entire

sinh is the same but with all plus signs (also for all z in C)

Question 6

Maclaurin series for $cosz$

Answer 6

Differentiate
$$ \sin{z} = z - \frac{z^3}{3!} + \frac{z^5}{5!} - \frac{z^7}{7!} + …$$
to get
$$ \cos{z} = 1 - \frac{z^2}{2!} + \frac{z^4}{4!} - \frac{z^6}{6!} + …$$

$\forall \ z \in \ C$ since cosz is entire

cosh is the same but with all plus signs (also for all z in C)