# Lecture - Analysis MT22, XIII

> Source: https://ollybritton.com/notes/uni/prelims/mt22/analysis-i/lectures/13/ · Updated: 2022-11-26 · Tags: uni, lecture

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

### Flashcards
What does it mean for $\sum^\infty_{k = 0} c_k x^k$ to be a real power series?::
$$
c_k, x \in \mathbb{R}
$$

What does it mean for $\sum^\infty_{k = 0} c_k z^k$ to be a complex power series?::
$$
c_k, z \in \mathbb{C}
$$

When considering power series, what is $0^0$ defined to be, by convention?::
$$
1
$$

How is the exponential function $e^z = \exp(z)$ defined in terms of a power series?::
$$
\sum^\infty_{k = 0} \frac{z^k}{k!}
$$

What’s a common technique for determining the radius of convergence of a power series?::
The ratio test.

How is the sine function $\sin z$ defined in terms of a power series?::
$$
\sin z = \sum_{k = 0}^\infty \frac{(-1)^k z^{2k+1}}{(2k+1)!}
$$

How is the cosine function $\cos z$ defined in terms of a power series?::
$$
\cos z = \sum^\infty_{k = 0} \frac{(-1)^k z^{2k}}{(2k)!}
$$

How is the hyperbolic sine function $\sinh z$ defined in terms of a power series?::
$$
\sinh z = \sum^\infty_{k = 0} \frac{z^{2k+1}}{(2k+1)!}
$$

How is the hyperbolic cosine function $\cosh z$ defined in terms of a power series?::
$$
\cosh z = \sum^\infty_{k = 0} \frac{z^{2k}}{(2k)!}
$$

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