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Chapter 4: Q. 31 (page 325)

Write out each sum in Exercises 31 in expanded form, and then calculate the value of the sum.

Short Answer

Expert verified

The expanded notation is $0 + \frac{1}{8} + \frac{1}{2} + \frac{9}{8} + 2 + \frac{25}{8}$ and their sum is $\frac{55}{8}$ or $6.875$ .

Step by step solution

01

Given information

The given summation is $\sum_{k = 0}^{5} {(\frac{1}{2} k)}^{2} (\frac{1}{2})$ .

02

Write the summation in expanded form and evaluate the sum.

Rewrite the given summation in expanded form.

$\begin{array}{rcl} \sum_{k = 0}^{5} {(\frac{1}{2} k)}^{2} (\frac{1}{2}) & = & {(\frac{1}{2} (0))}^{2} (\frac{1}{2}) + {(\frac{1}{2} (1))}^{2} (\frac{1}{2}) + {(\frac{1}{2} (2))}^{2} (\frac{1}{2}) + {(\frac{1}{2} (3))}^{2} (\frac{1}{2}) + {(\frac{1}{2} (4))}^{2} (\frac{1}{2}) + {(\frac{1}{2} (5))}^{2} (\frac{1}{2}) \\ = & 0 + \frac{1}{4} (\frac{1}{2}) + \frac{1}{2} + (\frac{9}{4}) (\frac{1}{2}) + (4) (\frac{1}{2}) + (\frac{25}{4}) (\frac{1}{2}) \\ = & 0 + \frac{1}{8} + \frac{1}{2} + \frac{9}{8} + 2 + \frac{25}{8} \\ = & \frac{35}{8} + \frac{5}{2} \\ = & \frac{35 + 20}{8} \\ = & \frac{55}{8} \\ = & 6.875 \end{array}$

03

Write the conclusion

The expanded form is $0 + \frac{1}{8} + \frac{1}{2} + \frac{9}{8} + 2 + \frac{25}{8}$ .

The sum is $\frac{55}{8}$ or $6.875$ .

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Most popular questions from this chapter

Use integration formulas to solve each integral in Exercises 21–62. 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.

$\int \frac{x + 1}{\sqrt{x}} d x$

Show by exhibiting a counterexample that, in general, $\int \frac{f (x)}{g (x)} d x \neq \frac{\int f (x) d x}{\int g (x) d x}$ . In other words, find two functions f and g such that the integral of their quotient is not equal to the quotient of their integrals.

Given a simple proof that $\sum_{k = 0}^{n} 3 a_{k} = 3 \sum_{k = 0}^{n} a_{k}$

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

$\int_{0}^{π} (\sin^{2} x + \cos^{2} x) d x$

Consider the general sigma notation $\sum_{k = m}^{n} a_{k}$ .What do we mean when we say that a_k is a function of k?

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