# Lecture - Analysis MT22, X

> Source: https://ollybritton.com/notes/uni/prelims/mt22/analysis-i/lectures/10/ · Updated: 2022-11-19 · Tags: uni, lecture

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

### Flashcards
Let $\sum_{k\ge 1} a_k$ be a series and $(s_n)$ be the series of $n$-th partial sums. What does it mean for the series to converge?::
$(s_n)$ converges.

Let $\sum_{k\ge 1} a_k$ be a series and $(s_n)$ be the series of $n$-th partial sums. How can you recover $a_n$ from $s_n$?::
$$
a_n = s_n - s_{n-1}
$$

Given that $a_n = s_n - s_{n-1}$ (where $s_n$ is the $n$-th partial sum), what is the necessary and sufficient condition for $a_n \ge 0$?::
$s_n$ is monotonic increasing.

What theorem/statement relates the convergence of $\sum_{k \ge 1} a_k$ and $a_k \to 0$?::
If $\sum_{k\ge 1} a_k$ converges, then $a_k \to 0$ as $k \to \infty$.

It is true that if $\sum_{k\ge 1} a_k$ converges, then $a_k \to 0$ as $k \to \infty$. What’s an example of the converse not being true?::
$$
a_k = \frac{1}{k}
$$

How can you prove the theorem that if $\sum_{k\ge 1} a_k$ converges, then $a_k \to 0$ as $k \to \infty$?::
Consider rewriting $a_n$ as $s_n - s_{n-1}$.

To prove that $\sum_{k\ge 1} \frac{1}{k}$ diverges, why is it enough to show $s_n = \sum^n_{k \ge 1} \frac{1}{k}$ is not Cauchy?::
Not being cauchy implies divergence.

$$$$ What $n$ and $m$ do you take in Cauchy’s convergence criterion for proving the divergence of the harmonic series?::
$$
n = 2^{p+1}\text{, }m = 2^p
$$

What’s the point of taking $n = 2^{p+1}\text{, }m = 2^p$ in Cauchy criterion for $s_n = \sum^n_{k \ge 1} \frac{1}{k}$?::
You can show that $|s_n - s_m|$ will always be greater than $\frac{1}{2}$.

If you have real series $\sum_{k\ge 1} a_k$ and $\sum_{k\ge 1} b_k$, what is the condition and statement for the simple comparison test?::
Condition:
$$
\exists c > 0 \text{ s.t } 0 \le a_k \le Cb_k \text{ for } k \ge 1
$$
Statement:
$$
\sum_{k\ge 1} b_k \text{ convergent} \implies \sum_{k \ge 1}a_k \text{ convergent}
$$

What does it mean for a series $\sum_{k \ge 1}a_k$ to be absolutely convergent?::
$$
\sum_{k \ge 1} |a_k| \text{ converges}
$$

What theorem relates absolute convergence and normal convergence for a real or complex series $\sum_{k \ge 1} a_k$?::
$$
\sum_{k \ge 1} |a_k| \text{ converges} \implies \sum_{k \ge 1} a_k \text{ converges}
$$

What does Cauchy’s criterion simplify into when considering the convergence of a series $\sum_{k \ge 1} a_k$?::
$$
\forall \epsilon > 0 \text{ } \exists N \text{ s.t. } n > m > N \implies |\sum^n_{k = m+1} a_k| < \epsilon
$$

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