Lecture - Analysis MT22, XI

Created: November 19, 2022

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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