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Chapter 4: Q. 40 (page 326)

Find a formula for each of the sums in Exercises 40, and then use these formulas to calculate each sum for $n = 100, 500$ and $1000$

Short Answer

Expert verified

The formula of the given summation is $\frac{n^{2} + 3 n + 5}{3 n^{2}}$ .

The sum when $n = 100$ is $0.3435$ .

The sum when $n = 500$ is $0.33534$ .

The sum when $n = 1000$ is $0.334335$ .

Step by step solution

01

Given information

The given summation is . ${鈭�}_{k = 1}^{n} \frac{k^{2} + k + 1}{n^{3}}$

02

Determine the formula for the given summation.

The sum can be written as:

${鈭�}_{k = 1}^{n} \frac{k^{2} + k + 1}{n^{3}} = \frac{1}{n^{3}} {鈭�}_{k = 1}^{n} (k^{2} + k + 1) = \frac{1}{n^{3}} ({鈭�}_{k = 1}^{n} k^{2} + {鈭�}_{k = 1}^{n} k + {鈭�}_{k = 1}^{n} 1) = \frac{1}{n^{3}} [(\frac{n (n + 1) (2 n + 1)}{6}) + \frac{n (n + 1)}{2} + n] = \frac{n^{2} + 3 n + 5}{3 n^{2}}$

03

Evaluate the sum for n=100,500 and 1000

Substitute $100$ for $n$ in $\frac{n^{2} + 3 n + 5}{3 n^{2}}$ .

role="math" localid="1649225056563" $\frac{{(100)}^{2} + 3 (100) + 5}{3 {(100)}^{2}} = 0.3435$

Substitute $500$ for $n$ in $\frac{n^{2} + 3 n + 5}{3 n^{2}}$ .

role="math" localid="1649225092265" $\frac{{(500)}^{2} + 3 (500) + 5}{3 {(500)}^{2}} = 0.33534$

Substitute $1000$ for $n$ in $\frac{n^{2} + 3 n + 5}{3 n^{2}}$ .

role="math" localid="1649225111347" $\frac{{(1000)}^{2} + 3 (1000) + 5}{3 {(1000)}^{2}} = 0.334335$

04

Write the conclusion

The formula is $\frac{n^{2} + 3 n + 5}{3 n^{2}}$ .

The sum when $n = 100, 500$ and $1000$ are $0.3435, 0.33534$ and $0.334335$ .

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Most popular questions from this chapter

Use integration formulas to solve each integral in Exercises 21鈥�62. You may have to use algebra, educated guess and-check, and/or recognize an integrand as the result of a product, quotient, or chain rule calculation. Check each of your answers by differentiating

$鈭� {(x^{3} + 4)}^{2} d x$ .

Write out all the integration formulas and rules that we know at this point.

If f is negative on [鈭�3, 2], is the definite integral ${鈭�}_{- 3}^{2} f (x) d x$ positive or negative? What about the definite integral 鈭� ${鈭�}_{- 3}^{2} f (x) d x$ ?

Show that $F (x) = \sin x - x \cos x + 2$ is an antiderivative of $f (x) = \sin x$ .

Given a simple proof that if n is a positive integer and c is any real number, then ${鈭�}_{k = 1}^{n} c = c n$

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