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Chapter 7: Q 80 (page 593)

Use the result of Exercise 79to approximate the square roots in Exercises 80-83. In each case, start with x0=1and stop when xk+1-xk<0.001.

2

Short Answer

Expert verified

The required approximate value is :-

2=1.414214.

Step by step solution

01

Step 1. Given Information

We have given the following square root :-

2.

We have to find the value of 2by using the recursion formula with initial approximation x0=1. Also we have to stop our process of find iteration whilexk+1-xk<0.001.

02

Step 2. To find the value of 2

In the previous example we find the following recursion formula :-

xk+1=12xk+axk, to find the approximate value of a.

Here a=2and x0=1, then the first iteration is :-

x1=121+21鈬�x1=12脳3鈬�x1=1.5

then second iteration is :-

x2=121.5+21.5鈬�x2=122.8334鈬�x2=1.41667

Then third iteration is :-

x3=121.41667+21.41667鈬�x3=1.414216

Then fourth iteration is :-

x4=121.414216+21.414216鈬�x4=1.414214

Here x4-x3=1.414214-1.414216=0.000002<0.001

Sox=1.414214is the required approximate value of2.

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Most popular questions from this chapter

Leila finds that there are more factors affecting the number of salmon that return to Redfish Lake than the dams: There are good years and bad years. These happen at random, but they are more or less cyclical, so she models the number of fish qkreturning each year as qk+1=(0.14(鈭�1)k+0.36)(qk+h), where h is the number of fish whose spawn she releases from the hatchery annually.

(a) Show that the sustained number of fish returning in even-numbered years approach approximately qe=3h鈭�k=1鈭�0.11k.

(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 qo=6111h鈭�k=1鈭�0.11k.

(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?

For a convergent series satisfying the conditions of the integral test, why is every remainder Rnpositive? How can Rnbe used along with the term Sn from the sequence of partial sums to understand the quality of the approximation Sn?

Explain why the integral test may be used to analyze the given series and then use the test to determine whether the series converges or diverges.

鈭�k=4鈭�鈥�2k鈭�4k2鈭�4k+3

In Exercises 48鈥�51 find all values of p so that the series converges.

鈭�k=1鈭�1(c+k)p,wherec>0

Determine whether the series 鈭�k=0鈭�2k+25k-1converges or diverges. Give the sum of the convergent series.
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