Lecture - Analysis MT22, XII

Created: November 26, 2022

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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