Lecture - Analysis MT22, XIV

Created: December 13, 2022

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 $ \vert z \vert < 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 $ \vert z \vert > 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 $ \vert z \vert = 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 $ \vert z \vert < \alpha$?

You also have to show it diverges for all $ \vert z \vert > \alpha$.

If you have a function $f(x) = \sum _ {k \ge 0} c _ k x^k$, for $ \vert x \vert < 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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