Homework

WeBWorK.

Get Your Grades

Lab

Welcome

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 \( 1 < x < 5 \).