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Calculus  II                                                             Quiz #8                                             April 28, 2003

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


(a)   1e^(-1),  2e^(-2),  3e^(-3), 4e^(-4),  ...  , n e^(-n),  ...

The nth term is n/e^x.  As n goes to infinity, this is an indeterminate form ∞/∞.
Using L'Hospital's rule:  Underscript[lim , n∞] n/e^x = Underscript[lim , n∞] 1/e^x = 0
Thus the sequence converges to 0.


(b)    {n/(5n + 15) + cos(π n)/n^2} _ (n = 1)^∞

Underscript[lim , n∞] n/(5n + 15) + cos(π n)/n^2 = Underscript[lim , n∞] n/(5n + 15) + Underscript[lim , n∞] cos(π n)/n^2 = 1/5 + 0 = 1/5
Thus the sequence converges to 1/5.



(c)    {(-n)^3/(6n^3 + n^2 + 3)} _ (n = 1)^∞
Underscript[lim , n∞] (-n)^3/(6n^3 + n^2 + 3) = Underscript[lim , n∞] -n^3/(6n^3 + n^2 + 3) = -1/6
Thus the sequence converges to -1/6

(2)   Consider the sequence  1/2^(1/2),     (1/21/3)^(1/2),    (1/21/31/4)^(1/2),    (1/21/31/41/5)^(1/2), ...
Decide if this sequence converges or diverges. Explain your reasoning.

Notice this is a decreasing sequence because
FormBox[RowBox[{a_ (n + 1)/a_n, =, RowBox[{(1/21/31/41/5 ... 1/n1/(n + 1))^(1/2)/(1/21/31/41/5 ... 1/2))/(1/21/31/41/5 ... 1/n)^(1/2), =, RowBox[{1/(n + 1)^(1/2), <, 1.}]}]}]}], TraditionalForm]

Moreover, the sequence is bounded below by 0.

As a decreasing sequence that is bounded below, it must converge.