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.

$S_{\infty} = \frac{1024}{5}$

Step by step solution

Given information:

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

Formulas:

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

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.

To find its sum:

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

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

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

Short Answer

Step by step solution

Given information:

Formulas:

To check whether the series converges or diverges:

To find its sum:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

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

Short Answer

Step by step solution

Given information:

Formulas:

To check whether the series converges or diverges:

To find its sum:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Pure Maths

Discrete Mathematics

Calculus

Probability and Statistics

Logic and Functions

Study anywhere. Anytime. Across all devices.

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