Notes - Analysis I MT22, Alternating series test
Flashcards
What’s the statement of the alternating series test for $\sum _ {k=1}^\infty (-1)^{k+1}a _ k$?
If $(a _ k)$ is a monotonically decreasing sequence and $a _ k \to 0$, then $\sum^\infty _ {k=1}(-1)^{k+1}a _ k$ converges.
What’s the basic idea when proving the alternating series test
If $(a _ k)$ is a monotonically decreasing sequence and $a _ k \to 0$, then $\sum^\infty _ {k=1}(-1)^{k+1}a _ k$ converges.
If $(a _ k)$ is a monotonically decreasing sequence and $a _ k \to 0$, then $\sum^\infty _ {k=1}(-1)^{k+1}a _ k$ converges.
Regroup terms of $s _ {2n}$ and $s _ {2n+1}$ to show they’re bounded monotonic subsequences.
When proving the alternating series test,
If $(a _ k)$ is a monotonically decreasing sequence and $a _ k \to 0$, then $\sum^\infty _ {k=1}(-1)^{k+1}a _ k$ converges.
You work with the partial sums $s _ {2n}$ and $s _ {2n+1}$. What two ways do you rearrange
\[\begin{aligned}
s_{2n} &= a_1 - a_2 + a_3 - a_4 + \ldots - a_{2n-2} + a_{2n-1} - a_{2n}
\end{aligned}\]
to show it’s both monotonically increasing and bounded above?
If $(a _ k)$ is a monotonically decreasing sequence and $a _ k \to 0$, then $\sum^\infty _ {k=1}(-1)^{k+1}a _ k$ converges.
When proving the alternating series test,
If $(a _ k)$ is a monotonically decreasing sequence and $a _ k \to 0$, then $\sum^\infty _ {k=1}(-1)^{k+1}a _ k$ converges.
You work with the partial sums $s _ {2n}$ and $s _ {2n+1}$. What two ways do you rearrange
\[\begin{aligned}
s_{2n+1} &= a_1 - a_2 + a_3 - a_4 + \ldots + a_{2n-1} - a_{2n} + a_{2n+1}
\end{aligned}\]
to show it’s both monotonically decreasing and bounded below?
If $(a _ k)$ is a monotonically decreasing sequence and $a _ k \to 0$, then $\sum^\infty _ {k=1}(-1)^{k+1}a _ k$ converges.
When proving the alternating series test,
If $(a _ k)$ is a monotonically decreasing sequence and $a _ k \to 0$, then $\sum^\infty _ {k=1}(-1)^{k+1}a _ k$ converges.
You end up showing that $s _ {2n} \to L$ and $s _ {2n+1} \to U$ both have limits. How do you show they’re equal?
If $(a _ k)$ is a monotonically decreasing sequence and $a _ k \to 0$, then $\sum^\infty _ {k=1}(-1)^{k+1}a _ k$ converges.
By the AOL,
\[\begin{aligned} U - L &= \lim_{n \to \infty} s_{2n+1} - s_{2n} \\\\ &= \lim_{n\to\infty} a_{2n+1} \\\\ &= 0 \end{aligned}\]