HW 7
Sec 9.1
28.
\( \displaystyle a_n = \frac{n+3}{n^2+5n+6}\).
Converges to 0.
32.
\( \displaystyle a_n = (-1)^n\bigg(1 - \frac{1}{n}\bigg) \).
Diverges.
60.
\( \displaystyle a_n = \frac{(-4)^n}{n!} \).
Converges to 0.
82.
\( \displaystyle a_n = \frac1{\sqrt{n^2-1} - \sqrt{n^2+n}} \).
Multiply the top and bottom by \(\sqrt{n^2-1} + \sqrt{n^2+n}\)
and simplify. The answer is \( -2 \).
106.
\( \displaystyle x_{n+1} = \max{x_n, \cos(n+1)} \).
This sequence is nondecreasing and bounded above by \( 1 \). Hence
it converges by Theorem 6.
Sec 9.2
12.
This converges to \( 17/2 \).
30.
This diverges since the terms don't go to 0.
48.
\begin{align*}
\sum_{n=0}^\infty (-1/2)^n (x-3)^n
&= \sum_{n=0}^\infty \bigg(\frac{-(x-3))}{2}\bigg)^n\\
&= 1/(1 - (3-x)/2)
\end{align*}
for \( 1 < x < 5 \).
|