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Exam 11: Infinite Sequences and Series

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Question 1

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Approximate the sum to the indicated accuracy. $\sum _ { n = 1 } ^ { \infty } \frac { 4 ( - 1 ) ^ { n - 1 } } { n ^ { 7 } }$ (five decimal places)

A) $6.97036$
B) $4.97036$
C) $7.97036$
D) $3.97036$
E) $5.97036$

F) C) and D)
G) B) and E)

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Question 2

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Find the radius of convergence and the interval of convergence of the power series. $\sum _ { n = 0 } ^ { \infty } \left( \frac { n x } { 6 } \right) ^ { n }$

A) $R = 0 , I = \{ 0 \}$
B) $R = \infty , I = ( - \infty , \infty )$
C) $R = 6 , I = [ - 6,6 ]$
D) $R = 6 , I = ( - 6,6 )$

E) All of the above
F) C) and D)

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Question 3

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Find the interval of convergence of the series. $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } x ^ { n } } { n + 3 }$

Evaluate the function $f ( x ) = \cos x$ by a Taylor polynomial of degree 4 centered at $a = 0$ , and $x = \frac { \pi } { 4 }$ . Select the correct answer.

Determine whether the sequence converges or diverges. If it converges, find the limit. $a _ { n } = e ^ { n / ( n + 6 ) }$

Determine whether the given series converges or diverges. If it converges, find its sum. $\sum _ { n = 1 } ^ { \infty } \left( 1 + \frac { 5 } { n } \right) ^ { n }$

A sequenceis $\left\{ a _ { n } \right\}$ defined recursively by the equation $a _ { n } = 0.5 \left( a _ { n - 1 } + a _ { n - 2 } \right)$ for $n \geq 3$ where $a _ { 1 } = 14 , a _ { 2 } = 14$ . Use your calculator to guess the limit of the sequence.

A sequenceis $\left\{ a _ { n } \right\}$ defined recursively by the equation $a _ { n } = 0.5 \left( a _ { n - 1 } + a _ { n - 2 } \right)$ for $n \geq 3$ where $a _ { 1 } = 14 , a _ { 2 } = 14$ . Use your calculator to guess the limit of the sequence.

Determine whether the geometric series converges or diverges. If it converges, find its sum. $\sum _ { n = 0 } ^ { \infty } 3 ^ { n } 4 ^ { - n + 1 }$ Select the correct answer.

$\text { Determine which series is convergent. }$

Determine whether the geometric series converges or diverges. If it converges, find its sum. $- \frac { 1 } { 5 } + \frac { 1 } { 25 } - \frac { 1 } { 125 } + \frac { 1 } { 625 } - \cdots$

Use the binomial series to expand the function as a power series. Find the radius of convergence. $\frac { x } { \sqrt { 16 + x ^ { 2 } } }$

Determine whether the series is absolutely convergent, conditionally convergent, or divergent. $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } \arctan n } { n ^ { 4 } }$

Find the value of the limit for the sequence given. $\left\{ \frac { 1 \cdot 9 \cdot 17 \cdots ( 7 n + 1 ) } { ( 7 n ) ^ { 2 } } \right\}$

For which positive integers $k$ is the series $\sum _ { n = 1 } ^ { \infty } \frac { ( n ! ) ^ { 5 } } { ( k n ) ! }$ convergent?

Let $a _ { k } = f ( k )$ , where $f$ is a continuous, positive, and decreasing function on $[ n , \infty )$ , and suppose that $\sum _ { k = 1 } ^ { \infty } a _ { k }$ is convergent. Defining $R _ { n } = S - S _ { n }$ , where $S = \sum _ { n = 1 } ^ { \infty } a _ { n }$ and $S _ { n } = \sum _ { k = 1 } ^ { n } a _ { k }$ , we have that $\int _ { n + 1 } ^ { \infty } f ( x ) d x \leq R _ { n } \leq \int _ { n } ^ { \infty } f ( x ) d x$ . Find the maximum error if the sum of the series $\sum _ { n = 1 } ^ { \infty } \frac { 3 } { n ^ { 2 } }$ is approximated by $S _ { 40 }$ .

Find the value of the limit for the sequence given. Select the correct answer. $\left\{ \frac { 1 \cdot 9 \cdot 17 \cdots ( 7 n + 1 ) } { ( 7 n ) ^ { 2 } } \right\}$

When money is spent on goods and services, those that receive the money also spend some of it. The people receiving some of the twice-spent money will spend some of that, and so on. Economists call this chain reaction the multiplier effect. In a hypothetical isolated community, the local government begins the process by spending $D$ dollars. Suppose that each recipient of spent money spends $100 c \%$ and saves $100 s \%$ of the money that he or she receives. The values $c$ and $s$ are called the marginal propensity to consume and the marginal propensity to save and, of course, $c + s = 1$ . The number $k = 1 / s$ is called the multiplier. What is the multiplier if the marginal propensity to consume is $90 \%$ ? Select the correct answer.

How many terms of the series $\sum _ { m = 2 } ^ { \infty } \frac { 12 } { 6 m ( \ln m ) ^ { 2 } }$ would you need to add to find its sum to within $0.02$ ?

Find a power series representation for the function. $f ( y ) = \ln \left( \frac { 11 + y } { 11 - y } \right)$