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Q 43.

Expert-verified

Calculus

Found in: Page 615

Book edition 1st

Author(s) Peter Kohn, Laura Taalman

Pages 1155 pages

ISBN 9781429241861

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Short Answer

Given that $a_{0} = - 3, a_{1} = 5, a_{2} = - 4, a_{3} = 2$ and ${\sum a_{k}}_{k = 2}^{\infty} = 7$ , find the value of ${\sum a_{k}}_{k = 0}^{\infty}$ .

The value of ${\sum a_{k}}_{k = 0}^{\infty} = 9$ .

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1. Given information.

Given that $a_{0} = - 3, a_{1} = 5, a_{2} = - 4, a_{3} = 2$ and ${\sum a_{k}}_{k = 2}^{\infty} = 7$ .

Step 2. Express the series in terms of known values.

Value of ${\sum a_{k}}_{k = 0}^{\infty}$ can be obtained as follows.

$\begin{array}{rcl} {\sum a_{k}}_{k = 0}^{\infty} & = & a_{0} + a_{1} + {\sum a_{k}}_{k = 2}^{\infty} \\ = & - 3 + 5 + 7 \\ = & 9 \end{array}$

Therefore, value of $\sum_{k = 0}^{\infty} a_{k}$ is 9.

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Examples: Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading.

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(Hint: Make a new recurrence by using two steps of the one given.)

(b) Show that the sustained number of fish returning in odd-numbered years approaches approximately $q_{o} = \frac{61}{11} h \sum_{k = 1}^{\infty} 0 . 11^{k} .$

(c) How should Leila choose h, the number of hatchery fish to breed in order to hold the minimum number of fish returning in each run near some constant P?

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Explain why, if n is an integer greater than 1, the series $\sum_{k = 1}^{\infty} \frac{1}{\sqrt[n]{k}}$ diverges.

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