# Lecture - Analysis MT22, XI

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

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

### Flashcards
$$\sum_{k \ge 1} (-1)^{k-1} u_k$$ What are the two conditions for a series of this form to converge, by the alternating series test?::
1. $u_k \to 0$ as $k \to \infty$
2. $u_k$ is monotonic decreasing

What’s an example of a series that is convergent, but not absolutely convergent?::
$$
\sum_{k \ge 1} \frac{(-1)^{k-1}}{k}
$$

What are the three conditions for a sequence $\sum_{k \ge 1} a_k$ converging if and only if $\sum_{k \ge 1} b_k$ is converging with the limit form of the comparison test?::
1. $b_k > 0$ for $k \ge 1$
2. $\frac{a_k}{b_k} \to L$ as $k \to \infty$
3. $L > 0$ but not infinite

If $a_k > 0$ for all $k \ge 1$ in $\sum_{k \ge 1}$ and $\frac{a_{k+1}}{a_k}$ tends towards a strictly positive limit, what can be said if $0 \le L < 1$?::
The series converges.

If $a_k > 0$ for all $k \ge 1$ in $\sum_{k \ge 1}$ and $\frac{a_{k+1}}{a_k}$ tends towards a strictly positive limit, what can be said if $L > 1$?::
The series diverges.

If $a_k > 0$ for all $k \ge 1$ in $\sum_{k \ge 1} a_k$ and $\frac{a_{k+1}}{a_k}$ tends towards a strictly positive limit, what can be said if $L = 1$?::
Nothing, the test is inconclusive.

Given $a_k = \frac{1}{k^2 + 2k}$, what $b_k$ would you choose in order to show that $\sum a_k$ converges via the limit form of the comparison test?::
$$
b_k = \frac{1}{k^2}
$$

Given $a_k = \frac{k^4 + 3k}{23k^7 - 11k + 8}$, what $b_k$ would you guess has the same growth rate in order to show that $\sum a_k$ converges via the limit form of the comparison test?::
$$
b_k = \frac{1}{k^3}
$$

Let $\sum^\infty_{k=1} a_k$ and $\sum^\infty_{k = 1} b_k$ be series and let $L = \lim_{k \to \infty} \frac{a_k}{b_k}$. What are the three cases of $L$ that need to be considered?::
- $L \in (0, \infty)$
- $L = 0$
- $L = \infty$

Let $\sum^\infty_{k=1} a_k$ and $\sum^\infty_{k = 1} b_k$ be series and let $L = \lim_{k \to \infty} \frac{a_k}{b_k}$. What is true about the convergence of $\sum^\infty_{k=1} a_k$ and $\sum^\infty_{k = 1} b_k$ if $L \in (0, \infty)$ and $b_k \ge 0$::
$$
\sum^\infty_{k=1} b_k \text{ converges} \iff \sum^\infty_{k = 1} a_k \text{ converges}
$$

Let $\sum^\infty_{k=1} a_k$ and $\sum^\infty_{k = 1} b_k$ be series and let $L = \lim_{k \to \infty} \frac{a_k}{b_k}$. What is true about the convergence of $\sum^\infty_{k=1} a_k$ and $\sum^\infty_{k = 1} b_k$ if $L = 0$?::
$$
\sum^\infty_{k=1} b_k \text{ converges} \implies \sum^\infty_{k = 1} a_k \text{ converges}
$$

Let $\sum^\infty_{k=1} a_k$ and $\sum^\infty_{k = 1} b_k$ be series and let $L = \lim_{k \to \infty} \frac{a_k}{b_k}$. What is true about the convergence of $\sum^\infty_{k=1} a_k$ and $\sum^\infty_{k = 1} b_k$ if $L = \infty$?::
$$
\sum^\infty_{k=1} b_k \text{ diverges} \implies \sum^\infty_{k = 1} a_k \text{ diverges}
$$

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