CPM Homework Help : APCALC Problem 12-69

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Home > APCALC > Chapter 12 > Lesson 12.2.1 > Problem 12-69

12-69.

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.

$\frac { 1 } { 3 } + \frac { 1 } { 6 } + \frac { 1 } { 11 } + \frac { 1 } { 18 } + \frac { 1 } { 27 } + \ldots$
Step (a):
$=\displaystyle \sum_{n=1}^\infty \frac{1}{n^2+2}$

$\frac{1}{2}+\frac{1}{9}+\frac{1}{28}+\frac{1}{65}+\frac{1}{126}+\frac{1}{217}+\ldots$
Step (b):
$=\displaystyle \sum_{n=1}^\infty \frac{1}{n^3+1}$

$\operatorname { ln } ( \frac { 1 } { 2 } ) + \operatorname { ln } ( \frac { 2 } { 3 } ) + \operatorname { ln } ( \frac { 3 } { 4 } ) +\ldots$
Step (c):
$= \ln(1)-\ln(2)+\ln(2)-\ln(3)+\ln(3)-\ln(4)+...$

$-2 + 1 -\frac { 2 } { 3 }+\frac { 1 } { 2 }-\frac { 2 } { 5 }+\ldots$
Hint (d):
Think of this as two separate series:
$\frac{-2}{1}+\frac{-2}{3}+\frac{-2}{5}+...\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+...$

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