Calculus II 
Quiz #8
April 24, 2002
R.  Hammack 
Score ______

(1) Decide if the following sequences converge or diverge. In the case of convergence, state the limit.

(a)   ln(1),  ln(1/2),  ln(1/3),  ln(1/4),  ...  , ln(1/n),  ...
This sequence diverges because ln(1/n) =  -ln(n)  approaches negative infinity as n approaches infinity.

(b)    [Graphics:Images/quiz8sol_gr_1.gif]

[Graphics:Images/quiz8sol_gr_2.gif][Graphics:Images/quiz8sol_gr_3.gif] = 2+0 = 2

Thus the sequence converges to 2.

(c)    [Graphics:Images/quiz8sol_gr_4.gif]

[Graphics:Images/quiz8sol_gr_5.gif][Graphics:Images/quiz8sol_gr_6.gif] [Graphics:Images/quiz8sol_gr_7.gif][Graphics:Images/quiz8sol_gr_8.gif] [Graphics:Images/quiz8sol_gr_9.gif][Graphics:Images/quiz8sol_gr_10.gif] [Graphics:Images/quiz8sol_gr_11.gif]cos(1/n) = cos 0 = 1
(form ∞ 0)         (form 0/0)

The sequence converges to 1.

Consider the sequence  [Graphics:Images/quiz8sol_gr_12.gif]    [Graphics:Images/quiz8sol_gr_13.gif]     [Graphics:Images/quiz8sol_gr_14.gif]    [Graphics:Images/quiz8sol_gr_15.gif] ...
Decide if this sequence converges or diverges. Explain your reasoning.

Notice that every term of this sequence is positive, and it decreases because any one term is obtained from the previous term by multiplying by a positive number [Graphics:Images/quiz8sol_gr_16.gif]that is less than 1. Therefore this sequence is a decreasing sequence that is bounded below by 0. Therefore, by Theorem 11.2.4, it converges.