CPM Homework Help : CALC Problem 12-70

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12-70.

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

$\displaystyle\sum _ { n = 1 } ^ { \infty } \frac { n \operatorname { ln } n } { 3 ^ { n } }$
Hint (a):
Use the Ratio Test.
Step 1(a):
$\lim\limits_{n\to\infty}\Bigg|\frac{\frac{(n+1)\ln(n+1)}{3^{n+1}}}{\frac{n\ln(n)}{3^n}}\Bigg|$
Step 2(a):
$\lim\limits_{n\to\infty}\Big|\frac{(n+1)\ln(n+1)}{3n\ln(n)}\Big|=?$

$\displaystyle\sum _ { k = 1 } ^ { \infty } \frac { 1 } { \sqrt { k ^ { 3 } + 2 } }$
Hint (b):
Compare this series to:
$\displaystyle \sum_{n=1}^\infty \frac{1}{n^{3/2}}$

$\displaystyle\sum _ { n = 2 } ^ { \infty } ( - 1 ) ^ { n + 1 } \frac { \operatorname { ln } n } { \operatorname { ln } n ^ { 2 } }$
Hint (c):
This is an alternating series, so what is
$\lim\limits_{n\to\infty}\frac{\ln(n)}{\ln(n^2)}=?$
More Help (c):
$\ln(n^2)=2\ln(n)$

$\displaystyle\sum _ { j = 1 } ^ { \infty } - \frac { j ^ { 2 } } { 2 ^ { j } }$
Hint (d):
Use the Ratio Test.

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