CPM Homework Help : APCALC Problem 12-8

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12-8.

Consider the infinite series below. For each series, decide if it diverges, converges conditionally, or converges absolutely and justify your conclusion. State the tests you used.

$\displaystyle \sum _ { n = 1 } ^ { \infty } ( 1 - \frac { 3 } { n } ) ^ { 3 n }$
Hint (a):
You should recognize the argument of the summation as representing a special value.

$\displaystyle \sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n + 1 } \frac { \operatorname { ln } ( n ) } { n }$
Hint (b):
This is an alternating series. What is:
$\lim \limits_{n\to\infty}\frac{\ln(n)}{n}?$
More Help (b):
Use l'Hôpital's Rule.

$\displaystyle \sum _ { k = 1 } ^ { \infty } \frac { \operatorname { ln } ( k ) } { k ^ { 3 } }$
Hint (c):
Compare this series to:
$\displaystyle \sum_{k=1}^\infty \frac{k}{k^3}$

$\displaystyle \sum _ { k = 5 } ^ { \infty } k ! e ^ { - k }$
Hint (d):
This is the same as the series below. Which is dominant when $k > 5$ , the numerator or the denominator?
$\displaystyle\sum_{k=5}^\infty\frac{k!}{e^k}$

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