# Lecture - Analysis MT22, XIV

> Source: https://ollybritton.com/notes/uni/prelims/mt22/analysis-i/lectures/14/ · Updated: 2022-12-13 · Tags: uni, lecture

- [Course - Analysis MT22](https://ollybritton.com/notes/uni/prelims/mt22/analysis-i/)

### Flashcards
If $\sum a_k \to L$ and $\sum b_k \to M$, what does $\sum a_k + b_k$ converge to?::
$$
L + M
$$

If $\sum a_k \to L$, then what does $\sum c a_k$ converge to?::
$$
cL
$$

If $z \in \mathbb{C}$, then what is $\cos z$ in terms of exponentials?::
$$
\cos z = \frac{1}{2}\left( e^{iz} + e^{-iz} \right)
$$

If $z \in \mathbb{C}$, then what is $\sin z$ in terms of exponentials?::
$$
\sin z = \frac{1}{2i}\left( e^{iz} - e^{-iz} \right)
$$

If $z \in \mathbb{C}$, then what is $\cosh z$ in terms of exponentials?::
$$
\cosh z = \frac{1}{2}\left( e^{z} + e^{-z} \right)
$$

If $z \in \mathbb{C}$, then what is $\sinh z$ in terms of exponentials?::
$$
\sinh z = \frac{1}{2}\left( e^{z} - e^{-z} \right)
$$

What is the definition of $R$, the radius of converge of a power series $\sum_{k \ge 0} c_k z^k$?::
$$
R = \begin{cases}\sup S \text{ where } S = \text{set}(|z| \text{ when } \sum c_k z^k \text{ converges} ) \\ \infty \text{ otherwise} \end{cases}
$$

If $\sum c_k z^k$ is a power series with radius of convergence $R$, then what can be said if $|z| < R$?::
$$
\sum_{k \ge 0} c_k z^k \text{ converges absolutely}
$$

If $\sum c_k z^k$ is a power series with radius of convergence $R$, then what can be said if $|z| > R$?::
$$
\sum_{k \ge 0} c_k z^k \text{ diverges}
$$

If $\sum c_k z^k$ is a power series with radius of convergence $R$, then what can be said if $|z| = R$?::
Nothing can be said about this in general, it has to be handled on a case-by-case basis.

What is the definition of the disc of convergence for a complex power series with radius of convergence $R$?::
$$
\\{ z \in \mathbb{C} \text{ } | \text{ } |z| < R \\}
$$

What is the definition of the interval of convergence for a real power series with radius of convergence $R$?::
$$
\\{ x \in \mathbb{R} \text{ } | \text{ } |z| < R \\}
$$

Why is it not possible to claim the radius of convergence of a power series is $\alpha$ if you show the power series converges for all $|z| < \alpha$?::
You also have to show it diverges for all $|z| > \alpha$.

If you have a function $f(x) = \sum_{k \ge 0} c_k x^k$, for $|x| < R$, then what does the differentiation theorem for real power series state?::
$$
f'(x) = \sum_{k \ge 1} kc_k x^{k-1} \text{ for }|x| < R
$$

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