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Chapter 4: Q. 14 (page 403)

The algebra of sums: Fill in the blanks to complete the sum rules that follow. You may assume that $a_{k}$ and $b_{k}$ are functions defined for nonnegative integers $k$ and that $c$ is any real number.
${鈭�}_{k = m}^{p - 1} a_{k} + {鈭�}_{k = p}^{n} a_{k} =$ ______

Short Answer

Expert verified

${鈭�}_{k = m}^{p - 1} a_{k} + {鈭�}_{k = p}^{n} a_{k} = {鈭�}_{k = m}^{n} a_{k}$ .

Step by step solution

01

Step 1. Given information

${鈭�}_{k = m}^{p - 1} a_{k} + {鈭�}_{k = p}^{n} a_{k}$ .

02

Step 2. First write the sigma notation in the expanded form.

${鈭�}_{k = m}^{p - 1} a_{k} + {鈭�}_{k = p}^{n} a_{k} = (a_{m} + a_{m + 1} + 鈰� + a_{p - 1}) + (a_{p} + a_{p + 1} + 鈰� + a_{n - 1} + a_{n}) = a_{m} + a_{m + 1} + 鈰� + a_{p - 1} + a_{p} + a_{p + 1} + 鈰� + a_{n - 1} + a_{n}$

Now write the obtained sequence in sigma notation.

$a_{m} + a_{m + 1} + 鈰� + a_{p - 1} + a_{p} + a_{p + 1} + 鈰� + a_{n - 1} + a_{n} = {鈭�}_{k = m}^{n} a_{k}$ .

where $m, n, p$ are any integers such that $0 鈮� m < p < n$ .

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Most popular questions from this chapter

Compare the definitions of the definite and indefinite integrals. List at least three things that are different about these mathematical objects.

Prove part (b) of theorem 4.4 in the case when n is even: if n is a positive even integer, then ${鈭�}_{k = 1}^{n} k = \frac{n (n + 1)}{2}$

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.

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

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

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

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