Sequences and Series

Write each of the arithmetic sequences in Exercises 47–50 in the form $\{c+dk{\}}_{k=0}^{\infty }$

$1,-1,\dots$

Write each of the arithmetic sequences in Exercises 47–50 in the form $\{c+dk{\}}_{k=0}^{\infty }$

$5,\pi ,\dots$

In Exercises 51–54 use the difference test in Theorem 7.6 to analyze the monotonicity of the given sequence.

$\left\{k^2-5k\right\}$

In Exercises 51–54 use the difference test in Theorem 7.6 to analyze the monotonicity of the given sequence.

$\left\{\sqrt{k}-\sqrt{k+1}\right\}$

In Exercises 51–54 use the difference test in Theorem 7.6 to analyze the monotonicity of the given sequence.

$\left\{\frac{k}{k+2}\right\}$

In Exercises 51–54 use the difference test in Theorem 7.6 to analyze the monotonicity of the given sequence.

$\left\{\frac{1}{k!}\right\}$

In Exercises 55– 58 use the ratio test in Theorem 7.6 to analyze the monotonicity of the given sequence.

$\left\{\frac{k^2}{k!}\right\}$

In Exercises 55– 58 use the ratio test in Theorem 7.6 to analyze the monotonicity of the given sequence.

$\left\{\sqrt{1-\frac{1}{k}}\right\}$

In Exercises 55– 58 use the ratio test in Theorem 7.6 to analyze the monotonicity of the given sequence.

$\left\{\frac{3^k}{2·4·6···\left(2k\right)}\right\}$

In Exercises 55– 58 use the ratio test in Theorem 7.6 to analyze the monotonicity of the given sequence.

$\left\{\frac{{\left(k!\right)}^2}{\left(2k\right)!}\right\}$

In Exercises 59–62 use the derivative test in Theorem 7.6 to analyze the monotonicity of the given sequence.

$\left\{\frac{\sqrt{k+1}}{k}\right\}$

In Exercises 59–62 use the derivative test in Theorem 7.6 to analyze the monotonicity of the given sequence.

$\left\{k^2-k\right\}$

In Exercises 59–62 use the derivative test in Theorem 7.6 to analyze the monotonicity of the given sequence.

$\left\{\frac{\sin k}{k}\right\}$

In Exercises 59–62 use the derivative test in Theorem 7.6 to analyze the monotonicity of the given sequence.

$\left\{\frac{k!}{\left(k+1\right)!}\right\}$

Determine whether the sequences in Exercises 63–74 are monotonic or not. Also determine whether the given sequence is bounded or unbounded.

$\left\{2-\frac{k-1}{10}\right\}$

Determine whether the sequences in Exercises 63–74 are monotonic or not. Also determine whether the given sequence is bounded or unbounded.

$\left\{\frac{1}{2k}\right\}$

Determine whether the sequences in Exercises 63–74 are monotonic or not. Also determine whether the given sequence is bounded or unbounded.

$a_k=\frac{{\left(-1\right)}^k}{k}$

Determine whether the sequences in Exercises 63–74 are monotonic or not. Also determine whether the given sequence is bounded or unbounded.

$a_k=\frac{k^2}{k!}$

Determine whether the sequences in Exercises 63–74 are monotonic or not. Also determine whether the given sequence is bounded or unbounded.

$\left\{1, -1, 1, -1, 1, ...\right\}$

Determine whether the sequences in Exercises 63–74 are monotonic or not. Also determine whether the given sequence is bounded or unbounded.

$\left\{1,\frac{3}{4},\frac{8}{9},\frac{15}{16},\frac{24}{25},...\right\}$

Determine whether the sequences in Exercises 63–74 are monotonic or not. Also determine whether the given sequence is bounded or unbounded. 

$\left\{\frac{\cos k}{k}\right\}$

Determine whether the sequences in Exercises 63–74 are monotonic or not. Also determine whether the given sequence is bounded or unbounded. 

$\left\{\frac{\cos \left(2\mathrm{\pi }k\right)}{k}\right\}$

Determine whether the sequences in Exercises 63–74 are monotonic or not. Also determine whether the given sequence is bounded or unbounded. 

$\left\{\frac{e^k}{k!}\right\}$

Determine whether the sequences in Exercises 63–74 are monotonic or not. Also determine whether the given sequence is bounded or unbounded. 

$\left\{\frac{\left(2k\right)!}{{\left(k!\right)}^2}\right\}$

Determine whether the sequences in Exercises 63–74 are monotonic or not. Also determine whether the given sequence is bounded or unbounded. 

$\left\{{\left(-1\right)}^{k}k\right\}$

Determine whether the sequences in Exercises 63–74 are monotonic or not. Also determine whether the given sequence is bounded or unbounded. 

$\left\{5\left(1-{\left(\frac{1}{10}\right)}^k\right)\right\}$

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

$f\left(x\right)=x^3-2, x_0=1$

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

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

Exercises 75–78 use Newton’s 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. $f\left(x\right)=e^x+\sin x,x_0=-2$

Exercises 75–78 use Newton’s method (see Example 8) to approximate a root for the given function with the specified value of $X_0$. Terminate your sequence when$\left|x_{n+1}-x_n\right|<0.001$

78. $f\left(x\right)=\sqrt{x+1}-\frac{1}{x},x_0=1$

Use Newton’s method to derive the recursion formula

$x_{k+1}=\frac{1}{2}\left(x_k+\frac{a}{x_k}\right)$

for approximating $\sqrt{a}$.

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 $\left|x_{k+1}-x_k\right|<0.001.$

$\sqrt{2}$

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 $\left|x_{k+1}-x_k\right|<0.001$.

81. $\sqrt{3}$

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. $\sqrt{4}$

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 $f\text{'}\left(x_0\right)=0$. 

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?

Use the result of Example $7$ to approximate the levels of the drug Excellent´e during the first week, assuming the dosages and decay rates in Exercises $86-88$.

$L_1=200, p=50$.

Use the result of Example $7$ to approximate the levels of the drug Excellent´e during the first week, assuming the dosages and decay rates in Exercises $86-88$.

$L_1=100, p=25$.

Many prescribed drugs must reach a “maintenance level” in the bloodstream to be effective. Say a person takes 300 milligrams of wonder drug Excellente´ per day and that whatever level of Excellente´ is in the bloodstream, p% is eliminated in one 24-hour period. 

Find the approximate level of drugs during the first week.

 Suppose you invest $100.00 in a bank that pays you 5% interest compounded annually. The balance in the account after k years is given by $a_k=100{(1-0.05)}^k$To the nearest cent, determine the first five terms of the sequence, starting at k = 0. What does k = 0 mean in practical terms ?Determine whether the sequence is bounded. Determine whether the sequence is increasing, decreasing, or not monotonic.

Prove that the ratio of successive terms of a nonzero geometric sequence is constant

 Prove that a sequence $\left\{a_k\right\}$ that is both increasing and decreasing is constant.

 Prove that every sequence of the form ${{\left\{a_k\right\}}_{k=n}}^{\infty }$can be rewritten as a sequence of the form ${{\left\{a_k\right\}}_{k=1}}^{\infty }$.

 Prove that if ${{\left\{a_k\right\}}_{k=1}}^{\infty }$ is a sequence of positive real numbers, then the sequence ${{\left\{S_n\right\}}_{n=1}}^{\infty }$,  where the sequence Sn = a1 + a2 +···+an, is an increasing sequence.

 Let $\left\{a_k\right\}$be a sequence. Prove Theorem 7.6 (a) along with the following variations:

(a) Show that when $a_{k-1}-a_k$≥ 0 for every k ≥ 1, the sequence is increasing.

(b) Show that when  $a_{k-1}-a_k$> 0 for every k ≥ 1, the sequence is strictly increasing.

(c) Show that when $a_{k-1}-a_k$≤ 0 for every k ≥ 1, the sequence is decreasing.

(d) Show that when $a_{k-1}-a_k$ < 0 for every k ≥ 1, the sequence is strictly decreasing. 

  Let $\left\{a_k\right\}$be a sequence of positive terms. Prove Theorem 7.6 (b) along with the following variations:

(a) Show that when $\frac{a_{k+1}}{a_k}\ge 1$≥ 1 for every k ≥ 1, the sequence is increasing.

(b) Show that when $\frac{a_{k+1}}{a_k}>1$ for every k ≥ 1, the sequence is strictly increasing.

(c) Show that when $\frac{a_{k+1}}{a_k}\le 1$for every k ≥ 1, the sequence is decreasing.

(d) Show that when $\frac{a_{k+1}}{a_k}<1$ for every k ≥ 1, the sequence is strictly decreasing.

Let a(x) be a differentiable function on the interval [1,∞), and let a_k = a(k) for every positive integer k. Prove Theorem 7.6 (c) along with the following variations: 

(a) Show that when a^'(x) ≥ 0f or x > 1,thesequence $\left\{a_k\right\}$ is increasing.

(b) Show that when a^'(x) > 0 for x > 1,thesequence $\left\{a_k\right\}$is strictly increasing.

 (c) Show that when a'(x) ≤ 0 for x > 1,thesequence $\left\{a_k\right\}$ is decreasing.

(d) Show that when a'(x) < 0, for x > 1, the sequence $\left\{a_k\right\}$ is strictly decreasing.

Consider the sequence \(\left \{ \frac{1}{k} \right \}_{k=0}^{\infty }\). The associated sequence \(\left \{ S_{n}\right \}_{n=0}^{\infty }\), where 

\(S_{n}=1+1+\frac{1}{2!}+...+\frac{1}{n!}\)

, is a sequence of sums. In Chapter \(8\) we will see that this sequence converges to the number \(e\). Evaluate \(S_n\) for \(n = 1, 2, 3, 10\). How close is \(S_10\) to \(e\)? 

Read the section and make your own summary of the material.