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Chapter 4: Q. 39 (page 326)

Find a formula for each of the sums in Exercises 39, and then use these formulas to calculate each sum for $n = 100, 500$ and $1000$
${鈭�}_{k = 1}^{n} \frac{k^{3} - 1}{n^{4}}$

Short Answer

Expert verified

The formula of the given summation is $\frac{(n - 1) (n^{2} + 3 n + 4)}{4 n^{3}}$ .

The sum when $n = 100$ is $0.255024$ . $0.255024$

The sum when localid="1649224503901" $n = 500$ is localid="1649224506104" $鈥� 0.251000992$ .

The sum whenlocalid="1649224508540" $n = 1000$ is $0.250500249$ . $0.250500249$ .

Step by step solution

01

Given information

The given summation is ${鈭�}_{k = 1}^{n} \frac{k^{3} - 1}{n^{4}}$ .

02

Determine the formula for the given summation.

The sum can be written as:

${鈭�}_{k = 1}^{n} \frac{k^{3} - 1}{n^{4}} = \frac{1}{n^{4}} {鈭�}_{k = 1}^{n} (k^{3} - 1) = \frac{1}{n^{4}} ({鈭�}_{k = 1}^{n} k^{3} - {鈭�}_{k = 1}^{n} 1) = \frac{1}{n^{4}} (\frac{n^{2} (n + 1)^{2}}{4} - n) = \frac{(n - 1) (n^{2} + 3 n + 4)}{4 n^{3}}$

03

Evaluate the sum for n=100,500 and 1000

Substitute $100$ for $n$ in $\frac{(n - 1) (n^{2} + 3 n + 4)}{4 n^{3}}$ .

localid="1649224281170" $\frac{(100 - 1) ({(100)}^{2} + 3 (100) + 4)}{4 {(100)}^{3}} = 0.255024$

Substitute $500$ for $n$ in $\frac{(n - 1) (n^{2} + 3 n + 4)}{4 n^{3}}$ .

localid="1649224305780" $\frac{(500 - 1) ({(500)}^{2} + 3 (500) + 4)}{4 {(500)}^{3}} = 鈥� 0.251000992$

Substitute $1000$ for $n$ in $\frac{(n - 1) (n^{2} + 3 n + 4)}{4 n^{3}}$ .

localid="1649224336994" $\frac{(1000 - 1) ({(1000)}^{2} + 3 (1000) + 4)}{4 {(1000)}^{3}} = 0.250500249$

04

Write the conclusion

The formula is $\frac{(n - 1) (n^{2} + 3 n + 4)}{4 n^{3}}$ .

The sum when $n = 100, 500$ and $1000$ are $0.255024, 0.251000992$ and $0.250500249$ .

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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.

${鈭�}_{2}^{4} 鈥� \frac{1}{4 鈭� 3 x} d x$

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

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

${鈭�}_{1}^{4} 鈥� \frac{2 x + 3}{x^{2} + 3 x + 4} d x$

Given a simple proof that ${鈭�}_{k = 0}^{n} 3 a_{k} = 3 {鈭�}_{k = 0}^{n} a_{k}$

Calculate the exact value of each definite integral in Exercises 47鈥�52 by using properties of definite integrals and the formulas in Theorem 4.13.

${鈭�}_{0}^{6} (3 x + 2) d x$

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