91影视

In Exercises 75鈥�78 use Newton鈥檚 method (see Example 8) to approximate a root for the given function with the specified value of $x_0.$Terminate your sequence when $\left|x_{n+1}-x_n\right|<0.001.$

$f\left(x\right)=x^3鈭�2,x_0=1$

Express each of the repeating decimals in Exercises 71鈥�78 as a geometric series and as the quotient of two integers reduced to lowest terms.

$1.272727...$

Prove Theorem 7.9. That is, let $a:[1,鈭�)鈫�\mathrm{鈩�}$be a continuous function and let $a_k=a\left(k\right)$for every $k鈭�{\mathrm{鈩�}}^{+}$. Show that if $\underset{x鈫�鈭�}{\lim }a\left(x\right)=L$, then $\left\{a_k\right\}鈫�L$

In Exercises $75-78$use Newton鈥檚 method (see Example $8$)to approximate a root for the given function with the specified value of $x_0$. Terminate your sequence when $\left|x_{n-1}-x_n\right|<0.001$.

$f\left(x\right)=e^x+\sin x,x_0=0$.

Prove that the converse of Theorem \(7.9\) is not true by finding a continuous function \(a:\left [ 1,\infty \right )\rightarrow R\) such that \(\lim_{x\rightarrow \infty }a\left \( x \right \)\) does not exist but \(\left \{ a\left \( x \right \) \right \}\) converges.

Express each of the repeating decimals in Exercises 71鈥�78 as a geometric series and as the quotient of two integers reduced to lowest terms.

$0.6345345...$

Express each of the repeating decimals in Exercises 71鈥�78 as a geometric series and as the quotient of two integers reduced to lowest terms.

$0.199999...$

Prove that if $\left\{a_k\right\}$is a sequence of nonzero terms with the property that $\underset{k鈫�鈭�}{\lim }{a}_k=鈭�$, then $\frac{1}{a_k}鈫�0$.

Exercises 75鈥�78 use Newton鈥檚 method (see Example 8) to approximate a root for the given function with the specified value of $x_0$. Terminate your sequence when $\left|x_{n+1}-x_n\right|<0.001$.

77. $f\left(x\right)=e^x+\sin x,x_0=-2$

Prove the statements about the convergence or divergence of sequences in Exercises 78鈥�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.

$k^p鈫�鈭�$