91Ó°ÊÓ

The figure shown is drawn recursively and then shaded. The largest square has side length \(1\) unit. A square whose side length is \(r%\) as long as the larger square is inscribed with one vertex on each edge of the larger square. This process is repeated recursively, resulting in shading as depicted in the figure. What is the area of the shaded portion of the picture?

Prove the statements about the convergence or divergence of sequences in Exercises 78–83, referring to theorems in the section as necessary. For each of these statements, assume that r is a real number and p is a positive real number.

If$\left|r\right|<1,$then the sequence$r^k→0$

Use the result of Exercise 79 to approximate the square roots in Exercises 80–83. In each case, start with $x_0=1$and stop when $\left|x_{k+1}−x_k\right|<0.001$.

82.$\sqrt{4}$

The figure shown is called a Sierpinski triangle. It may be constructed recursively as follows: We start with a large black equilateral triangle. Every time we see a black equilateral triangle, we inscribe a white equilateral triangle with each vertex at the midpoint of a side of the black triangle. If the side of the largest triangle is \(1\) unit and this process is repeated recursively, what is the area of the white-shaded region?

Use the result of Exercise 79 to approximate the square roots in Exercises 80–83. In each case, start with $x_0=1$and stop when $\left|x_{k+1}−x_k\right|<0.001$.

83.$\sqrt{101}$

Explain why Newton’s method will fail if you choose a value of $x_0$ for which $f\text{'}\left(x_0\right)=0$.

Prove that if $\underset{k→∞}{\lim }{a}_k=L,$then localid="1649337757642" $\underset{k→∞}{\lim }{a}_{k+1}=L$

Prove Theorem 7.24 (a). That is, show that if c is a real number and$\underset{k=1}{\overset{∞}{∑}}{a}_k$ is a convergent series, then $\underset{k=1}{\overset{∞}{∑}}c{a}_k=c\underset{k=1}{\overset{∞}{∑}}{a}_k$.

Newton's approach will also not work if $\left|x_{k+1}-x_k\right|,$the difference between subsequent approximations, does not diminish as $k$rises.

(a) Demonstrate that when you select $x_0=0$, this occurs for the function $f\left(x\right)=\sqrt[3]{x-2}$

(b) What does $f\left(x\right)=\sqrt[3]{x-2}$ have as its root?

Prove that if $\underset{k=1}{\overset{∞}{∑}}{a}_k$converges to L and $\underset{k=1}{\overset{∞}{∑}}{b}_k$converges to M , then the series$\underset{k=1}{\overset{∞}{∑}}\left(a_k+b_k\right)=L+M$.