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10-102.

Determine if the following series converge or diverge. State the test you used.

$\displaystyle\sum _ { n = 1 } ^ { \infty } n ( \frac { 2 } { 3 } ) ^ { n }$
Hint (a):
The Ratio Test can be used.

$\displaystyle\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n + \sqrt { n } }$
Hint (b):
$n+\sqrt{n}\le 2n$

$\displaystyle\sum _ { n = 1 } ^ { \infty } \operatorname { ln } ( \frac { n } { n + 1 } )$
Hint (c):
$\ln\Big(\frac{n}{n+1}\Big)=\ln(n)-\ln(n+1)$
More Help (c):
Write out some terms of the series. You should notice that this is a telescoping series.

$\displaystyle\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ( \operatorname { ln } n ) ^ { 2 } }$
Hint (d):
What is the value of the first term of the series?

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