CPM Homework Help : APCALC Problem 12-22

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12-22.

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

$\displaystyle \sum _ { k = 0 } ^ { \infty } \frac { k ^ { 2 } } { \sqrt { k ^ { 5 } + 7 } }$
Hint (a):
Compare the argument of this summation to:
$\frac{k^2}{\sqrt{k^5}}=k^{-1/2}$

$\displaystyle \sum _ { j = 2 } ^ { \infty } \frac { 1 } { j \sqrt { \operatorname { ln } ( j ) } }$
Hint (b):
Use the Integral Test. To evaluate the integral, let $u = \ln(j)$ .

$\displaystyle \sum _ { j = 1 } ^ { \infty } j ^ { 10 } ( \frac { 9 } { 10 } ) ^ { j }$
Hint (c):
Use the Ratio Test.
Step (c):
$\lim \limits_{j\to\infty}\Bigg|\frac{j^{10}\Big(\frac{9}{10}\Big)^{j+1}}{j^{10}\Big(\frac{9}{10}\Big)^j}\Bigg|$

$\displaystyle \sum _ { n = 4 } ^ { \infty } \frac { 1 } { \sqrt { n ^ { 5 } - 1000 } }$
Hint (d):
Compare this series to:
$\displaystyle \sum_{n=5}^{\infty}\frac{1}{n^2}$

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