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Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 12: Q. 12 (page 839)

Determine whether the infinite geometric series:
$256 - 64 + 16 - 4 + . . .$
converges or diverges. If it converges, find its sum.

Short Answer

Expert verified

$256 - 64 + 16 - 4 + . . .$ converges.

$S_{鈭�} = \frac{1024}{5}$

Step by step solution

01

Given information:

Consider the series, $256 - 64 + 16 - 4 + . . .$

02

Formulas:

$(1) C o m m o n r a t i o : r = \frac{a_{n}}{a_{n - 1}} (2) S_{鈭�} = \frac{a_{1}}{1 - r}$

03

To check whether the series converges or diverges:

Given Series, $256 - 64 + 16 - 4 + . . .$

Rewriting the sum of geometric series: $256 + (- 64) + 16 + (- 4) + . . .$

Here, $a_{1} = 256, a_{2} = - 64, a_{3} = 16, a_{4} = - 4, . . .$

$r = \frac{- 64}{256} = \frac{- 1}{4} r = \frac{16}{- 64} = \frac{- 1}{4} r = \frac{- 4}{16} = \frac{- 1}{4}$

Common Ratio, $r = \frac{- 1}{4} = - 0.75 < 1$

Since $|r| < 1$ , the series converges.

04

To find its sum:

Here, $a_{1} = 256, r = \frac{- 1}{4}$

$S_{鈭�} = \frac{256}{1 - (\frac{- 1}{4})} [U s e f o r m u l a (2)] = \frac{256}{1 + \frac{1}{4}} = \frac{256}{\frac{4 + 1}{4}} = 256 脳 \frac{4}{5} = \frac{1024}{5}$

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