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Chapter 4: Q. 24 (page 325)

Write each of the sums in Exercises 21鈥�28 in sigma notation. Identify $m, n, a_{k}$ in each problem.
$\frac{1}{4} + \frac{1}{9} + \frac{1}{16} + \frac{1}{25} + \frac{1}{36} + \frac{1}{49} + \frac{1}{64}$

Short Answer

Expert verified

The sigma notation is ${鈭�}_{k = 2}^{8} \frac{1}{k^{2}} a n d m = 2, n = 8, a_{k} = \frac{1}{k^{2}}$

Step by step solution

01

Step 1. Given Information

The given sum is $\frac{1}{4} + \frac{1}{9} + \frac{1}{16} + \frac{1}{25} + \frac{1}{36} + \frac{1}{49} + \frac{1}{64}$

02

Step 2. Explanation

The sum in sigma notation is written as,

$\frac{1}{4} + \frac{1}{9} + \frac{1}{16} + \frac{1}{25} + \frac{1}{36} + \frac{1}{49} + \frac{1}{64} = \frac{1}{2^{2}} + \frac{1}{3^{2}} + \frac{1}{4^{2}} + \frac{1}{5^{2}} + \frac{1}{6^{2}} + \frac{1}{7^{2}} + \frac{1}{8^{2}} = {鈭�}_{k = 2}^{8} \frac{1}{k^{2}} w h e r e, m = 2, n = 8, a_{k} = \frac{1}{k^{2}}$

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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.

${鈭�}_{鈭� 3}^{2} 鈥� \frac{1}{(x + 5)^{2}} d x$

Consider the region between f and g on [0, 4] as in the

graph next at the left. (a) Draw the rectangles of the left-

sum approximation for the area of this region, with n = 8.

Then (b) express the area of the region with definite

integrals that do not involve absolute values.

Prove Theorem 4.13(c): For any real numbers a and b, ${鈭�}_{a}^{b} x^{2} d x = \frac{1}{3} (b^{3} - a^{3}) .$ Use the proof of Theorem 4.13(a) as a guide.

Sum and constant-multiple rules: State the sum and constant-multiple rules for (a) derivatives and (b) limits.

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

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

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