# Lecture - Analysis MT22, XV

> Source: https://ollybritton.com/notes/uni/prelims/mt22/analysis-i/lectures/15/ · Updated: 2022-12-13 · Tags: uni, lecture

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

### Flashcards
What is the power series definition of $\log(1 + x)$?::
$$
\sum_{k \ge 1} \frac{(-1)^{k-1} x^k}{k}
$$

What’s the trick to proving $\exp(\log(1+x)) = 1+x$ for all $x$?::
Show $g(x) = \frac{\exp(\log(1+x))}{1 + x}$ has a constant derivative.

If a function has a zero derivative everywhere, what is true?::
The function is constant.

When a function has a $0$ derivative, what is often convinient to say about $g(x)$ when working with power series?::
$$
g(x) = g(0)
$$

When working with proofs like $\exp(a + b) = \exp(a)\exp(b)$ using the differentiation theorem, what is often the first step?::
Fixing one of the arguments to be a constant real number and the other to be $x$.

What is the precise definition of $\pi$?::
$$
\pi = \inf \\{x \in \mathbb{R} \text{ } | \text{ } x > 0 \text{ and } \cos(x/2) = 0\\}
$$

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