CPM Homework Help : CALC Problem 9-45

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9-45.

Determine whether each of the infinite series converges or diverges. Explain each answer briefly. If the series converges, find its exact sum.

$\displaystyle\sum _ { k = 1 } ^ { \infty } 1$
Hint (a):
What is $1 + 1 + 1 + 1 + ...$ ?

$\displaystyle\sum _ { j = 2 } ^ { \infty } 8 ( \frac { 3 } { 4 } ) ^ { j }$
Hint (b):
This is a geometric sequence with a ratio of $\frac{3}{4}$ .

$\displaystyle\sum _ { k = 0 } ^ { \infty } 20 ( 0.4 ) ^ { k }$
Hint (c):
This is also a geometric sequence. What is the ratio? (See part (b).)

$\displaystyle\sum _ { n = 1 } ^ { \infty } \frac { 3 } { n ( n + 1 ) }$
Steps (d):
This can be written as a telescoping sum:
$S=3\bigg(\frac{1}{2}+\frac{1}{6}+\frac{1}{12}+...\bigg)$
$\text{ }=3\bigg(\Big(1-\frac{1}{2}+\Big(\frac{1}{2}-\frac{1}{3}\Big)+\Big(\frac{1}{3}-\frac{1}{4}\Big)+...\bigg)$

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