# Lecture - Analysis MT22, XII

> Source: https://ollybritton.com/notes/uni/prelims/mt22/analysis-i/lectures/12/ · Updated: 2022-11-26 · Tags: uni, lecture

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

### Flashcards
Given $s_n = \sum^n_{k = 1} f(k)$, what are the three conditions for the integral test for $f:[1, \infty) \to \mathbb{R}$ to be applicable?::
- $f$ is non-negative.
- $f$ is decreasing.
- $\int^{k+1}_{k} f(x) \text{d}x$ exists for each $k \ge 1$.

What’s a simpler (but stronger) statement instead of that $\int^{k+1}_{k} f(x) \text{d}x$ exists for each $k \ge 1$ when using the integral test?::
$f(x)$ is continuous.

Let $S_n = \sum^n_{k=1} f(k)$ and $I_n = \int^n_1 f(x) \text{d}x$ for $f$ satisfying all the necessary conditions. What is the first statement of the integral test theorem, about $\sigma_n = S_n - I_n$?::
$\sigma_n$ converges to some $\sigma$, and $0 \le \sigma \le f(1)$.

Let $S_n = \sum^n_{k=1} f(k)$ and $I_n = \int^n_1 f(x) \text{d}x$ for $f$ satisfying all the necessary conditions. What is the second statement of the integral test theorem, about the relationship between $S_n$ and $I_n$?::
$$
\sum_{k=1}^n f(k) \text{ converges} \iff \int^n_1 f(x) \text{d}x \text{ converges}
$$

What’s the definition of Euler’s constant $\gamma$?::
$$
\gamma = \lim_{n \to \infty} \left( \left( \sum^{n}_{k=1} \frac{1}{k} \right) - \log n\right)
$$

In what sense is Euler’s constant $\gamma$ “not special”?::
The difference between any sum and integral of a suitable function will converge.

What’s a common technique when considering the limit of alternating sequences like
$$
1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \ldots
$$
?::
Consider $s_{2n}$ and $s_{2n-1}$ since you’ll know for certain if the last term is negative or positive.

What does it mean for $\sum b_k$ to be a rearrangement of $\sum a_k$?::
There exists a bijection $g : \mathbb{N} \to \mathbb{N}$ s.t. $b_k = a_{g(k)}$.

Why are absolutely convergent series “robust”?::
Any rearrangement converges to the same value.

Why are conditionally convergent series “delicate”?::
Rearragements might diverge or converge to different limits.

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