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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. 3 (page 839)

Write out the sum. Evaluate the sum.
${鈭�}_{k = 1}^{3} {(- 1)}^{k + 1} (\frac{k + 1}{k^{2}})$

Short Answer

Expert verified

The value of the given sum is $\frac{61}{36} .$

Step by step solution

01

Given information:

Consider , ${鈭�}_{k = 1}^{3} {(- 1)}^{k + 1} (\frac{k + 1}{k^{2}})$

To expand the sum and then to find the value of it.

02

To expand the given sum:

${鈭�}_{k = 1}^{3} {(- 1)}^{k + 1} (\frac{k + 1}{k^{2}}) = [{(- 1)}^{1 + 1} (\frac{1 + 1}{1^{2}})] + [{(- 1)}^{2 + 1} (\frac{2 + 1}{2^{2}})] + [{(- 1)}^{3 + 1} (\frac{3 + 1}{3^{2}})]$

03

Simplification:

= [{(- 1)}^{2} (\frac{2}{1})] + [{(- 1)}^{3} (\frac{3}{4})] + [{(- 1)}^{4} (\frac{4}{9})] = 2 - \frac{3}{4} + \frac{4}{9} = \frac{(2 脳 36) - (3 脳 9) + (4 脳 4)}{36} = \frac{72 - 27 + 16}{36} = \frac{61}{36}

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Most popular questions from this chapter

In Problems 51鈥�66, determine whether each infinite geometric series converges or diverges. If it converges, find its sum.

$9 + 12 + 16 + \frac{64}{3} + . . . .$

In Problems 61鈥�70, express each sum using summation notation.

$1 - \frac{1}{3} + \frac{1}{9} - \frac{1}{27} + . . . . . . + {(- 1)}^{6} \frac{1}{3^{6}}$

In Problems 51鈥�66, determine whether each infinite geometric series converges or diverges. If it converges, find its sum.

${鈭�}_{k = 1}^{鈭�} 3 {(\frac{3}{2})}^{k - 1}$

If $|r| < 1$ , the sum of the geometric series a

${鈭�}_{k = 1}^{鈭�} a r^{k - 1}$ is ________ .

Fibonacci Sequence: Let

$u_{n} = \frac{{(1 + \sqrt{5})}^{n} - {(1 - \sqrt{5})}^{n}}{2^{n} \sqrt{5}}$

define the nth term of a sequence.

(a) Show that u₁= 1 and u₂ = 1.

(b) Show that u_n+2 = u_n+1 + u_n.

(c) Draw the conclusion that {u_n} is the Fibonacci sequence.

See all solutions

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