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Chapter 1: Q1P (page 35)

Prove theorem $14.3$ . Hint: Group the terms in the error as role="math" localid="1657423688910" $(a_{n + 1} + a_{n + 2}) + (a_{n + 3} + a_{n + 4}) + 鈰�$ to show that the error has the same sign as role="math" localid="1657423950271" $a_{n + 1} .$ Then group them asrole="math" localid="1657423791335" $a_{n + 1 + (a_{n + 2} + a_{n + 3}) + (a_{n + 4} + a_{n + 5}) + 鈰�}$ to show that the error has magnitude less than $| a_{n + 1} | .$

Short Answer

Expert verified

The alternating sequence converges.

$|S - (a_{1} + a_{2} + 鈥� + a_{n})| 鈮� |a_{n + 1}|$

Step by step solution

01

Given Information

The Properties of alternating sequence.

$S = {鈭�}_{n = 0}^{鈭�} a_{n} |a_{n + 1}| < |a_{n}| \lim_{n 鈫� 鈭�} a_{n} = 0$

02

Definition of Alternating series

A mathematical series in which the terms are alternately positive and negative is known as an alternating series.

03

Solve for convergence of given series

Use properties mentioned below:

$S = {鈭�}_{n = 0}^{鈭�} a_{n} |a_{n + 1}| < |a_{n}| \lim_{n 鈫� 鈭�} a_{n} = 0$

Solve left hand side of the theorem.

$S - (a_{1} + a_{2} + 鈥� + a_{n}) = a_{n + 1} + a_{n + 2} + a_{n + 3} + a_{n + 4} 鈰� = (a_{n + 1} + a_{n + 2}) + (a_{n + 3} + a_{n + 4}) + 鈥� a_{n + 1} S - (a_{1} + a_{2} + 鈥� + a_{n}) = a_{n + 1} + (a_{n + 2} + a_{n + 3}) + (a_{n + 4} + a_{n + 5}) + 鈥� |S - (a_{1} + a_{2} + 鈥� + a_{n})| 鈮� |a_{n + 1}|$

Hence, the alternating sequence converges.

$|S - (a_{1} + a_{2} + 鈥� + a_{n})| 鈮� |a_{n + 1}|$

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Most popular questions from this chapter

Connect the midpoints of the sides of an equilateral triangle to form 4 smaller equilateral triangles. Leave the middle small triangle blank, but for each of the other 3 small triangles, draw lines connecting the midpoints of the sides to create 4 tiny triangles. Again leave each middle tiny triangle blank and draw the lines to divide the others into 4 parts. Find the infinite series for the total area left blank if this process is continued indefinitely. (Suggestion: Let the area of the original triangle be 1; then the area of the first blank triangle is 1/4.) Sum the series to find the total area left blank. Is the answer what you expect? Hint: What is the 鈥渁rea鈥� of a straight line? (Comment: You have constructed a fractal called the Sierpinski gasket. A fractal has the property that a magnified view of a small part of it looks very much like the original.)

In the bouncing ball example above, find the height of the tenth rebound, and the distance traveled by the ball after it touches the ground the tenth time. Compare this distance with the total distance traveled.

By the method used to obtain (12.5)[which is the series(13.1)below], verify each of the other series (13.2)to (13.5)below.

A periodic modulated (AM) radio signal has the form $y = (A + B s i n 2 蟺 f t) s i n 2 蟺 f_{e} (t - \frac{x}{v})$ .

The factor $s i n 2 蟺 f_{c} (t - x / v)$ is called the carrier wave; it has a very high frequency (called radio frequency; $f_{c}$ is of the order of $10^{6}$ cycles per second). The amplitude of the carrier wave is $(A + B s i n 2 蟺 f t)$ . This amplitude varies with time-hence the term "amplitude modulation"-with the much smaller frequency of the sound being transmitted (called audio frequency;fis of the order of $10^{2}$ cycles per second). In order to see the general appearance of such a wave, use the following simple but unrealistic data to sketch a graph of y as a function oftfor x = 0 over two periods of the amplitude function: $A = 3, B = 1, f = 1, f_{c} = 20$ . Using trigonometric formulas, show thatycan be written as a sum of three waves of frequencies $f_{c} f_{c} + f$ , and $f_{c} - f$ ; the first of these is the carrier wave and the other two are called side bands.

Find the work done by the force is $F = (2 x y - 3) i + x^{2} j$ in moving an object from (1,0) to (0,1) long each of the three paths shown:

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