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Chapter 4: Q. 41 (page 326)

Write each expression in Exercises 41 in one sigma notation (with some extra terms added to or subtracted from the sum, as necessary).
$2 {鈭�}_{k = 0}^{100} a_{k} - {鈭�}_{k = 3}^{101} a_{k}$

Short Answer

Expert verified

The expression in one sigma is $2 a_{0} + 2 a_{1} + 2 a_{2} - a_{101} + {鈭�}_{k = 3}^{100} a_{k}$ .

Step by step solution

01

Given information

The given summation is $2 {鈭�}_{k = 0}^{100} a_{k} - {鈭�}_{k = 3}^{101} a_{k}$ .

02

Determine the expression is one sigma.

Rewrite the given summation in one sigma form.

$2 {鈭�}_{k = 0}^{100} a_{k} - {鈭�}_{k = 3}^{101} a_{k} = 2 (a_{0} + a_{1} + a_{2} + {鈭�}_{k = 3}^{100} a_{k}) - {鈭�}_{k = 3}^{101} a_{k} = 2 a_{0} + 2 a_{1} + 2 a_{2} + 2 {鈭�}_{k = 3}^{100} a_{k} - ({鈭�}_{k = 3}^{100} a_{k} + a_{101}) = 2 a_{0} + 2 a_{1} + 2 a_{2} + {鈭�}_{k = 3}^{100} a_{k} - a_{101} = 2 a_{0} + 2 a_{1} + 2 a_{2} - a_{101} + {鈭�}_{k = 3}^{100} a_{k}$

03

Write the conclusion

The one sigma notation is $2 a_{0} + 2 a_{1} + 2 a_{2} - a_{101} + {鈭�}_{k = 3}^{100} a_{k}$ .

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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

$鈭� {(x^{3} + 4)}^{2} d x$ .

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

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

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.

Determine which of the limit of sums in Exercises 47鈥�52 are infinite and which are finite. For each limit of sums that is finite, compute its value.

$\lim_{n 鈫� 鈭�} {鈭�}_{k = 1}^{n} \frac{k^{2} + k + 1}{n}$

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