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Calculus  II                                           Quiz #8                     April 23, 2003

Name_________________            R.  Hammack                  Score ______
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(1) Decide if the following sequences converge or diverge. In the case of convergence, state the limit.

(a)   { 4sec(5/n)   } _ (n = 1)^∞

Underscript[lim , n∞] 4sec(5/n) = 4sec(0) = 4  (CONVERGES)

(b)    {(n + ln(n))/(3n + 1)} _ (n = 1)^∞

Underscript[lim , n∞] (n + ln(n))/(3n + 1) = Underscript[lim , n∞] (1 + 1/n)/3 = 1/3  (CONVERGES)


(c)    {(-1)^n/( n^2 ) + 1/n !} _ (n = 1)^∞

Underscript[lim , n∞] ((-1)^n/( n^2 ) + 1/n !) = 0 + 0 = 0  (CONVERGES)

(2)   Consider the sequence defined recursively as a_1 = 1,   and   a_ (n + 1) = a_n/(1 + a_n).

(a) Explain why this series converges.

Each term is positive, so the sequence is bounded below by 0.
Also, the sequence is decreasing, since each term equals the previous term divided by a number that is bigger than 1.
Since the sequence is decreasing and bounded below, it CONVERGES.

(b)  Find the limit.
We know the limit exists by part a. Call the limit L, so Underscript[lim , n∞] a_n = L.
Now,  a_ (n + 1) = a_n/(1 + a_n),
so   Underscript[lim , n∞] a_ (n + 1) = Underscript[lim , n∞] a_n/(1 + a_n),
which gives   L =L/(1 + L).
Solving,   L(1+L)=L
L^2 + L = L
L^2 = 0
Thus the limit is L = 0