CPM Homework Help : CALC Problem 9-114

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9-114.

Consider the infinite series below. For each, decide if it converges or diverges and justify your conclusion. If the series converges, find its sum.

Hint:

For an infinite geometric series:

$\frac { 2 } { 3 } + \frac { 1 } { 3 } + \frac { 1 } { 6 } + \frac { 1 } { 12 } + \dots$
Hint (a):
This is a geometric series with $r =\frac{1}{2}$ .

$0.9 + 0.99 + 0.999 + 0.9999 + ...$
Hint (b):
Think of this as:
$0.9 + (0.9 + 0.09) + (0.9 + 0.09 + 0.009) + ...$
What is happening to the value of each term?

$0.1 + 0.12 + 0.123 + 0.1234 + ...$
Hint (c):
See the hint in part (b).

$256 − 128 + 64 − 32 + ...$
Hint (d):
This is a geometric series with $r =-\frac{1}{2}$ .

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