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Chapter 4: Q. 23 (page 325)

Write each of the sums in Exercises 21鈥�28 in sigma notation. Identify $m, n, a_{k}$ in each problem.
$3 + \frac{4}{8} + \frac{5}{27} + \frac{6}{64} + \frac{7}{125}$

Short Answer

Expert verified

The sigma notation is ${鈭�}_{k = 1}^{5} \frac{k + 2}{k^{3}}$

Step by step solution

01

Step 1. Given Information

The given sum is $3 + \frac{4}{8} + \frac{5}{27} + \frac{6}{64} + \frac{7}{125}$

02

Step 2. Explanation

The sum in sigma notation is written as,

$3 + \frac{4}{8} + \frac{5}{27} + \frac{6}{64} + \frac{7}{125} = \frac{1 + 2}{1^{3}} + \frac{2 + 2}{2^{3}} + \frac{3 + 2}{3^{3}} + \frac{4 + 2}{4^{3}} + \frac{5 + 2}{5^{3}} = {鈭�}_{k = 1}^{5} \frac{k + 2}{k^{3}} W h e r e, m = 1, n = 5, a_{k} = \frac{k + 2}{k^{3}}$

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Most popular questions from this chapter

Use the Fundamental Theorem of Calculus to find the exact values of the given definite integrals. Use a graph to check your answer.

${鈭�}_{0}^{1} 鈥� \frac{1}{1 + x^{2}} d x$

Use the Fundamental Theorem of Calculus to find the exact values of the given definite integrals. Use a graph to check your answer.

${鈭�}_{0}^{蟺} 鈥� (\sin^{2} 鈦� x + \cos^{2} 鈦� x) d x$

Show that $F (x) = \frac{1}{2} e^{(x^{2})} (x^{2} - 1)$ is an anti-derivative of $f (x) = x^{3} e^{(x^{2})} .$

Without calculating any sums or definite integrals, determine the values of the described quantities. (Hint: Sketch graphs first.)

(a) The signed area between the graph of f(x) = cos x and the x-axis on [鈭捪€, 蟺].

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(c) The area of the region between the graphs of f(x) = $\sqrt{4 鈭� x^{2}} and g (x) = 鈭� \sqrt{4 鈭� x^{2}} on [鈭� 2, 2] .$

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