Exam 11: Infinite Sequences and Series

A rubber ball is dropped from a height of 8 m onto a flat surface.Each time the ball hits the surface, it rebounds to 50% of its previous height.Find the total distance the ball travels.

Free

(Multiple Choice)

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Question 1

Correct Answer:

Verified

B

Suppose that the radius of convergence of the power series $\sum _ { n = 0 } ^ { \infty } c _ { n } x ^ { n }$ is 9 . What is the radius of convergence of the power series $\sum _ { n = 0 } ^ { \infty } c _ { n } x ^ { 2 n }$ .

Free

(Multiple Choice)

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(75)

Question 2

Correct Answer:

Verified

B

Write the first five terms of the sequence $\left\{ a _ { n } \right\}$ whose $n ^ { \text {th } }$ term is given. $a _ { n } = \frac { n + 7 } { 6 n - 1 }$

Free

(Short Answer)

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Question 3

Correct Answer:

Verified

$\frac { 8 } { 5 } , \frac { 9 } { 11 } , \frac { 10 } { 17 } , \frac { 11 } { 23 } , \frac { 12 } { 29 }$

Find all positive values of $u$ for which the series $\sum _ { m - 1 } ^ { \infty } 6 u ^ { \ln 7 m }$ converges.

(Multiple Choice)

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Question 4

Use series to approximate the definite integral to within the indicated accuracy. $\int _ { 0 } ^ { 05 } x ^ { 2 } e ^ { - x ^ { 2 } } d x \quad \mid \text { error } \mid < 0.001$

(Multiple Choice)

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Question 5

Test the series for convergence or divergence. $\sum _ { n = 0 } ^ { \infty } \frac { 1 } { \sqrt { n ^ { 5 } + 8 } }$

(Short Answer)

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Question 6

Determine whether the series is convergent or divergent. $\sum _ { n = 1 } ^ { \infty } \frac { 9 ^ { n } } { n ! n }$

(Short Answer)

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Question 7

Test the series for convergence or divergence. $\sum _ { n = 2 } ^ { \infty } ( - 1 ) ^ { n } \frac { n } { 5 \ln n }$

(Short Answer)

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Question 8

Find all positive values of $u$ for which the series $\sum _ { m = 1 } ^ { \infty } 6 u ^ { m } 7 m$ converges.

(Multiple Choice)

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Question 9

Test the series for convergence or divergence. $\sum _ { m = 1 } ^ { \infty } ( - 4 ) ^ { m } \frac { \ln m } { \sqrt { m } }$

(Short Answer)

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Question 10

Find the sum of the series. $\frac { 2 } { 1 \cdot 3 } - \frac { 2 ^ { 2 } } { 2 \cdot 3 ^ { 2 } } + \frac { 2 ^ { 3 } } { 3 \cdot 3 ^ { 3 } } - \frac { 2 ^ { 4 } } { 4 \cdot 3 ^ { 4 } } + \ldots$

(Multiple Choice)

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Question 11

Find the sum of the series. $\sum _ { n = 0 } ^ { \infty } \frac { 2 ^ { n } } { 3 ^ { n } n ! }$

(Short Answer)

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Question 12

Determine which one of the p-series below is divergent.

(Multiple Choice)

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Question 13

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

(Multiple Choice)

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(80)

Question 14

Determine whether the sequence defined by $a _ { n } = \frac { n ^ { 2 } - 5 } { 6 n ^ { 2 } + 1 }$ converges or diverges. If it converges, find its limit.

(Multiple Choice)

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(75)

Question 15

Find the values of p for which the series is convergent. $\sum _ { n = 2 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { \left( \ln \left( n ^ { 6 } \right) \right) ^ { p } }$

(Multiple Choice)

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Question 16

Determine whether the series is convergent or divergent. $\sum _ { n = 1 } ^ { \infty } \left( \frac { \ln \left( n ^ { 6 } \right) } { n } \right) ^ { n }$

(Short Answer)

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Question 17

Test the series for convergence or divergence. $\sum _ { k = 5 } ^ { \infty } \frac { 5 } { k ( \ln k ) ^ { 7 } }$

(Short Answer)

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Question 18

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

(Multiple Choice)

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Question 19

Determine whether the series is convergent or divergent. $\sum _ { n = 1 } ^ { \infty } \frac { ( n ! ) ^ { 4 } } { ( 7 n ) ! }$

(Short Answer)

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Question 20

A rubber ball is dropped from a height of 8 m onto a flat surface.Each time the ball hits the surface, it rebounds to 50% of its previous height.Find the total distance the ball travels.

Suppose that the radius of convergence of the power series $\sum _ { n = 0 } ^ { \infty } c _ { n } x ^ { n }$ is 9 . What is the radius of convergence of the power series $\sum _ { n = 0 } ^ { \infty } c _ { n } x ^ { 2 n }$ .

Write the first five terms of the sequence $\left\{ a _ { n } \right\}$ whose $n ^ { \text {th } }$ term is given. $a _ { n } = \frac { n + 7 } { 6 n - 1 }$

Find all positive values of $u$ for which the series $\sum _ { m - 1 } ^ { \infty } 6 u ^ { \ln 7 m }$ converges.

Use series to approximate the definite integral to within the indicated accuracy. $\int _ { 0 } ^ { 05 } x ^ { 2 } e ^ { - x ^ { 2 } } d x \quad \mid \text { error } \mid < 0.001$

Test the series for convergence or divergence. $\sum _ { n = 0 } ^ { \infty } \frac { 1 } { \sqrt { n ^ { 5 } + 8 } }$

Determine whether the series is convergent or divergent. $\sum _ { n = 1 } ^ { \infty } \frac { 9 ^ { n } } { n ! n }$

Test the series for convergence or divergence. $\sum _ { n = 2 } ^ { \infty } ( - 1 ) ^ { n } \frac { n } { 5 \ln n }$

Find all positive values of $u$ for which the series $\sum _ { m = 1 } ^ { \infty } 6 u ^ { m } 7 m$ converges.

Test the series for convergence or divergence. $\sum _ { m = 1 } ^ { \infty } ( - 4 ) ^ { m } \frac { \ln m } { \sqrt { m } }$

Find the sum of the series. $\frac { 2 } { 1 \cdot 3 } - \frac { 2 ^ { 2 } } { 2 \cdot 3 ^ { 2 } } + \frac { 2 ^ { 3 } } { 3 \cdot 3 ^ { 3 } } - \frac { 2 ^ { 4 } } { 4 \cdot 3 ^ { 4 } } + \ldots$

Find the sum of the series. $\sum _ { n = 0 } ^ { \infty } \frac { 2 ^ { n } } { 3 ^ { n } n ! }$

Determine which one of the p-series below is divergent.

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

Determine whether the sequence defined by $a _ { n } = \frac { n ^ { 2 } - 5 } { 6 n ^ { 2 } + 1 }$ converges or diverges. If it converges, find its limit.

Find the values of p for which the series is convergent. $\sum _ { n = 2 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { \left( \ln \left( n ^ { 6 } \right) \right) ^ { p } }$

Determine whether the series is convergent or divergent. $\sum _ { n = 1 } ^ { \infty } \left( \frac { \ln \left( n ^ { 6 } \right) } { n } \right) ^ { n }$

Test the series for convergence or divergence. $\sum _ { k = 5 } ^ { \infty } \frac { 5 } { k ( \ln k ) ^ { 7 } }$

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

Determine whether the series is convergent or divergent. $\sum _ { n = 1 } ^ { \infty } \frac { ( n ! ) ^ { 4 } } { ( 7 n ) ! }$

Functions and Models

Limits and Derivatives

Differentiation Rules

Applications of Differentiation

Integrals

Applications of Integration

Techniques of Integration

Further Applications of Integration

Differential Equations

Parametric Equations and Polar Coordinates

Vectors and the Geometry of Space

Vector Functions

Partial Derivatives

Multiple Integrals

Vector Calculus

Second-Order Differential Equations

Filters

Exam 11: Infinite Sequences and Series

A rubber ball is dropped from a height of 8 m onto a flat surface.Each time the ball hits the surface, it rebounds to 50% of its previous height.Find the total distance the ball travels.

Suppose that the radius of convergence of the power series ∑n=0∞cnxn\sum _ { n = 0 } ^ { \infty } c _ { n } x ^ { n }∑n=0∞​cn​xn is 9 . What is the radius of convergence of the power series ∑n=0∞cnx2n\sum _ { n = 0 } ^ { \infty } c _ { n } x ^ { 2 n }∑n=0∞​cn​x2n .

Write the first five terms of the sequence {an}\left\{ a _ { n } \right\}{an​} whose nth n ^ { \text {th } }nth term is given. an=n+76n−1a _ { n } = \frac { n + 7 } { 6 n - 1 }an​=6n−1n+7​

Find all positive values of uuu for which the series ∑m−1∞6uln⁡7m\sum _ { m - 1 } ^ { \infty } 6 u ^ { \ln 7 m }∑m−1∞​6uln7m converges.

Use series to approximate the definite integral to within the indicated accuracy. ∫005x2e−x2dx∣ error ∣<0.001\int _ { 0 } ^ { 05 } x ^ { 2 } e ^ { - x ^ { 2 } } d x \quad \mid \text { error } \mid < 0.001∫005​x2e−x2dx∣ error ∣<0.001

Test the series for convergence or divergence. ∑n=0∞1n5+8\sum _ { n = 0 } ^ { \infty } \frac { 1 } { \sqrt { n ^ { 5 } + 8 } }∑n=0∞​n5+8​1​

Determine whether the series is convergent or divergent. ∑n=1∞9nn!n\sum _ { n = 1 } ^ { \infty } \frac { 9 ^ { n } } { n ! n }∑n=1∞​n!n9n​

Test the series for convergence or divergence. ∑n=2∞(−1)nn5ln⁡n\sum _ { n = 2 } ^ { \infty } ( - 1 ) ^ { n } \frac { n } { 5 \ln n }∑n=2∞​(−1)n5lnnn​

Find all positive values of uuu for which the series ∑m=1∞6um7m\sum _ { m = 1 } ^ { \infty } 6 u ^ { m } 7 m∑m=1∞​6um7m converges.

Test the series for convergence or divergence. ∑m=1∞(−4)mln⁡mm\sum _ { m = 1 } ^ { \infty } ( - 4 ) ^ { m } \frac { \ln m } { \sqrt { m } }∑m=1∞​(−4)mm​lnm​

Find the sum of the series. 21⋅3−222⋅32+233⋅33−244⋅34+…\frac { 2 } { 1 \cdot 3 } - \frac { 2 ^ { 2 } } { 2 \cdot 3 ^ { 2 } } + \frac { 2 ^ { 3 } } { 3 \cdot 3 ^ { 3 } } - \frac { 2 ^ { 4 } } { 4 \cdot 3 ^ { 4 } } + \ldots1⋅32​−2⋅3222​+3⋅3323​−4⋅3424​+…

Find the sum of the series. ∑n=0∞2n3nn!\sum _ { n = 0 } ^ { \infty } \frac { 2 ^ { n } } { 3 ^ { n } n ! }∑n=0∞​3nn!2n​

Determine which one of the p-series below is divergent.

Find the value of the limit for the sequence given. {1⋅9⋅17⋯(7n+1)(7n)2}\left\{ \frac { 1 \cdot 9 \cdot 17 \cdots ( 7 n + 1 ) } { ( 7 n ) ^ { 2 } } \right\}{(7n)21⋅9⋅17⋯(7n+1)​}

Determine whether the sequence defined by an=n2−56n2+1a _ { n } = \frac { n ^ { 2 } - 5 } { 6 n ^ { 2 } + 1 }an​=6n2+1n2−5​ converges or diverges. If it converges, find its limit.

Find the values of p for which the series is convergent. ∑n=2∞(−1)n(ln⁡(n6))p\sum _ { n = 2 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { \left( \ln \left( n ^ { 6 } \right) \right) ^ { p } }∑n=2∞​(ln(n6))p(−1)n​

Determine whether the series is convergent or divergent. ∑n=1∞(ln⁡(n6)n)n\sum _ { n = 1 } ^ { \infty } \left( \frac { \ln \left( n ^ { 6 } \right) } { n } \right) ^ { n }∑n=1∞​(nln(n6)​)n

Test the series for convergence or divergence. ∑k=5∞5k(ln⁡k)7\sum _ { k = 5 } ^ { \infty } \frac { 5 } { k ( \ln k ) ^ { 7 } }∑k=5∞​k(lnk)75​

Find the radius of convergence and the interval of convergence of the power series. ∑n=0∞(nx6)n\sum _ { n = 0 } ^ { \infty } \left( \frac { n x } { 6 } \right) ^ { n }∑n=0∞​(6nx​)n

Determine whether the series is convergent or divergent. ∑n=1∞(n!)4(7n)!\sum _ { n = 1 } ^ { \infty } \frac { ( n ! ) ^ { 4 } } { ( 7 n ) ! }∑n=1∞​(7n)!(n!)4​

Functions and Models

Limits and Derivatives

Differentiation Rules

Applications of Differentiation

Integrals

Applications of Integration

Techniques of Integration

Further Applications of Integration

Differential Equations

Parametric Equations and Polar Coordinates

Vectors and the Geometry of Space

Vector Functions

Partial Derivatives

Multiple Integrals

Vector Calculus

Second-Order Differential Equations

Filters

Suppose that the radius of convergence of the power series $\sum _ { n = 0 } ^ { \infty } c _ { n } x ^ { n }$ is 9 . What is the radius of convergence of the power series $\sum _ { n = 0 } ^ { \infty } c _ { n } x ^ { 2 n }$ .

Write the first five terms of the sequence $\left\{ a _ { n } \right\}$ whose $n ^ { \text {th } }$ term is given. $a _ { n } = \frac { n + 7 } { 6 n - 1 }$

Find all positive values of $u$ for which the series $\sum _ { m - 1 } ^ { \infty } 6 u ^ { \ln 7 m }$ converges.

Use series to approximate the definite integral to within the indicated accuracy. $\int _ { 0 } ^ { 05 } x ^ { 2 } e ^ { - x ^ { 2 } } d x \quad \mid \text { error } \mid < 0.001$

Test the series for convergence or divergence. $\sum _ { n = 0 } ^ { \infty } \frac { 1 } { \sqrt { n ^ { 5 } + 8 } }$

Determine whether the series is convergent or divergent. $\sum _ { n = 1 } ^ { \infty } \frac { 9 ^ { n } } { n ! n }$

Test the series for convergence or divergence. $\sum _ { n = 2 } ^ { \infty } ( - 1 ) ^ { n } \frac { n } { 5 \ln n }$

Find all positive values of $u$ for which the series $\sum _ { m = 1 } ^ { \infty } 6 u ^ { m } 7 m$ converges.

Test the series for convergence or divergence. $\sum _ { m = 1 } ^ { \infty } ( - 4 ) ^ { m } \frac { \ln m } { \sqrt { m } }$

Find the sum of the series. $\frac { 2 } { 1 \cdot 3 } - \frac { 2 ^ { 2 } } { 2 \cdot 3 ^ { 2 } } + \frac { 2 ^ { 3 } } { 3 \cdot 3 ^ { 3 } } - \frac { 2 ^ { 4 } } { 4 \cdot 3 ^ { 4 } } + \ldots$

Find the sum of the series. $\sum _ { n = 0 } ^ { \infty } \frac { 2 ^ { n } } { 3 ^ { n } n ! }$

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

Determine whether the sequence defined by $a _ { n } = \frac { n ^ { 2 } - 5 } { 6 n ^ { 2 } + 1 }$ converges or diverges. If it converges, find its limit.

Find the values of p for which the series is convergent. $\sum _ { n = 2 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { \left( \ln \left( n ^ { 6 } \right) \right) ^ { p } }$

Determine whether the series is convergent or divergent. $\sum _ { n = 1 } ^ { \infty } \left( \frac { \ln \left( n ^ { 6 } \right) } { n } \right) ^ { n }$

Test the series for convergence or divergence. $\sum _ { k = 5 } ^ { \infty } \frac { 5 } { k ( \ln k ) ^ { 7 } }$

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

Determine whether the series is convergent or divergent. $\sum _ { n = 1 } ^ { \infty } \frac { ( n ! ) ^ { 4 } } { ( 7 n ) ! }$