수열의 수렴과 발산

Use the Squeeze Theorem to find the limit of each of the following sequences.

${a}_{n}=\sin n\sin (1/n)$

For the following sequences, plot the first $25$ terms of the sequence and state whether the graphical evidence suggests that the sequence converges or diverges.

[T] ${a}_{n}=\sin n$

For the following sequences, plot the first $25$ terms of the sequence and state whether the graphical evidence suggests that the sequence converges or diverges.

[T] ${a}_{n}=\cos n$

Newton’s method seeks to approximate a solution $f(x)=0$ that starts with an initial approximation ${x}_{0}$ and successively defines a sequence ${x}_{n+1}={x}_{n}-\frac{f({x}_{n})}{{f}^{'}({x}_{n})}.$ For the given choice of $f$ and ${x}_{0},$ write out the formula for ${x}_{n+1}.$ If the sequence appears to converge, give an exact formula for the solution $x,$ then identify the limit $x$ accurate to four decimal places and the smallest $n$ such that ${x}_{n}$ agrees with $x$ up to four decimal places.

[T] $f(x)={x}^{2}-2,$ ${x}_{0}=1$

Newton’s method seeks to approximate a solution $f(x)=0$ that starts with an initial approximation ${x}_{0}$ and successively defines a sequence ${x}_{n+1}={x}_{n}-\frac{f({x}_{n})}{{f}^{'}({x}_{n})}.$ For the given choice of $f$ and ${x}_{0},$ write out the formula for ${x}_{n+1}.$ If the sequence appears to converge, give an exact formula for the solution $x,$ then identify the limit $x$ accurate to four decimal places and the smallest $n$ such that ${x}_{n}$ agrees with $x$ up to four decimal places.

[T] $f(x)={e}^{x}-2,$ ${x}_{0}=1$

Newton’s method seeks to approximate a solution $f(x)=0$ that starts with an initial approximation ${x}_{0}$ and successively defines a sequence ${x}_{n+1}={x}_{n}-\frac{f({x}_{n})}{{f}^{'}({x}_{n})}.$ For the given choice of $f$ and ${x}_{0},$ write out the formula for ${x}_{n+1}.$ If the sequence appears to converge, give an exact formula for the solution $x,$ then identify the limit $x$ accurate to four decimal places and the smallest $n$ such that ${x}_{n}$ agrees with $x$ up to four decimal places.

[T] $f(x)=\ln x-1,$ ${x}_{0}=2$

[T] Consider a series combining geometric growth and arithmetic decrease. Let ${a}_{1}=1.$ Fix $a>1$ and $0<b<a.$ Set ${a}_{n+1}=a.{a}_{n}-b.$ Find a formula for ${a}_{n+1}$ in terms of ${a}^{n},$ $a,$ and $b$ and a relationship between $a$ and $b$ such that ${a}_{n}$ converges.

[T] To find an approximation for $\pi ,$ set ${a}_{0}=\sqrt{2+1},$ ${a}_{1}=\sqrt{2+{a}_{0}},$ and, in general, ${a}_{n+1}=\sqrt{2+{a}_{n}}.$ Finally, set ${p}_{n}={3.2}^{n+1}\sqrt{2-{a}_{n}}.$ Find the first ten terms of ${p}_{n}$ and compare the values to $\pi .$

[T] Starting with $K=16,807$ and $M=2,147,483,647,$ using ten different starting values of ${a}_{1},$ compute sequences of bits ${b}_{n}$ up to $n=1000,$ and compare their averages to ten such sequences generated by a random bit generator.

[T] Find the first $1000$ digits of $\pi$ using either a computer program or Internet resource. Create a bit sequence ${b}_{n}$ by letting ${b}_{n}=1$ if the $n\text{th}$ digit of $\pi$ is odd and ${b}_{n}=0$ if the $n\text{th}$ digit of $\pi$ is even. Compute the average value of ${b}_{n}$ and the average value of ${d}_{n}=|{b}_{n+1}-{b}_{n}|,$ $n=1,...,999.$ Does the sequence ${b}_{n}$ appear random? Do the differences between successive elements of ${b}_{n}$ appear random?

In Exercises 5–8, give the first five terms of the given sequence.

$\displaystyle \{a_n\} = \left\{\frac{4^n}{(n+1)!}\right\}$

In Exercises 5–8, give the first five terms of the given sequence.

$\displaystyle \{b_n\} = \left\{\left(-\frac32\right)^n\right\}$

In Exercises 5–8, give the first five terms of the given sequence.

$\displaystyle \{c_n\} = \left\{-\frac{n^{n+1}}{n+2}\right\}$

In Exercises 5–8, give the first five terms of the given sequence.

$\displaystyle \{d_n\} = \left\{\frac{1}{\sqrt{5}}\left(\left(\frac{1+\sqrt{5}}{2}\right)^n-\left(\frac{1-\sqrt{5}}{2}\right)^n\right)\right\}$

In Exercises 9–12, determine the $n^\text{th}$ term of the given sequence.

4, 7, 10, 13, 16, $\ldots$

In Exercises 9–12, determine the $n^\text{th}$ term of the given sequence.

$\displaystyle 3,\ -\frac32,\ \frac34,\ -\frac38,\ \ldots$

In Exercises 9–12, determine the $n^\text{th}$ term of the given sequence.

$10,\ 20,\ 40,\ 80,\ 160,\ \ldots$

In Exercises 9–12, determine the $n^\text{th}$ term of the given sequence.

$\displaystyle 1, 1,\ \frac12,\ \frac16,\ \frac1{24},\ \frac1{120},\ \ldots$

In Exercises 13–16, use the following information to determine the limit of the given sequences.

box90pt$\displaystyle \{a_n\} = \left\{\frac{2^n-20}{2^n}\right\}$; $\displaystyle \lim_{n\to\infty} a_n = 1$

box90pt$\displaystyle \{b_n\} = \left\{\left(1+\frac{2}{n}\right)^n\right\}$; $\displaystyle \lim_{n\to\infty} b_n = e^2$

box90pt$\displaystyle \{c_n\} = \left\{\sin (3/n)\right\}$; $\displaystyle \lim_{n\to\infty} c_n = 0$

$\displaystyle \{d_n\} = \left\{ \frac{2^n-20}{7\cdot2^n} \right\}$

In Exercises 13–16, use the following information to determine the limit of the given sequences.

box90pt$\displaystyle \{a_n\} = \left\{\frac{2^n-20}{2^n}\right\}$; $\displaystyle \lim_{n\to\infty} a_n = 1$

box90pt$\displaystyle \{b_n\} = \left\{\left(1+\frac{2}{n}\right)^n\right\}$; $\displaystyle \lim_{n\to\infty} b_n = e^2$

box90pt$\displaystyle \{c_n\} = \left\{\sin (3/n)\right\}$; $\displaystyle \lim_{n\to\infty} c_n = 0$

$\displaystyle \{d_n\} = \left\{ 3b_n-a_n \right\}$