# Lecture - Analysis MT22, XV

### 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 $\pi$?

\[\pi = \inf \\{x \in \mathbb{R} \text{ } | \text{ } x > 0 \text{ and } \cos(x/2) = 0\\}\]