Exam 11: Infinite Sequences and Series

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 $\sum _ { n = 0 } ^ { \infty } c _ { n } x ^ { 2 n }$ .

(Multiple Choice)

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Question 61

Determine whether the sequence convergent or divergent. $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 2 } - 6 n + 10 }$

(Short Answer)

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Question 62

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

(Short Answer)

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Question 63

Determine whether the sequence defined by $a _ { n } = \frac { \sin 2 n } { 9 n }$ converges or diverges. If it converges, find its limit.

(Short Answer)

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Question 64

Test the series for convergence or divergence. $\sum _ { m = 1 } ^ { \infty } \frac { ( - 6 ) ^ { m + 1 } } { 4 ^ { 8 m } }$

(Short Answer)

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Question 65

Which of the given series are absolutely convergent?

(Multiple Choice)

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Question 66

Determine whether the series is absolutely convergent, conditionally convergent, or divergent. Select the correct answer. $\sum _ { n = 1 } ^ { \infty } \left( \frac { 4 n ^ { 2 } + 3 } { 3 n ^ { 2 } + 4 } \right) ^ { n }$

(Multiple Choice)

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Question 67

Use series to evaluate the limit correct to three decimal places. $\lim _ { x \rightarrow 0 } \frac { 7 x - \tan ^ { - 1 } 7 x } { x ^ { 3 } }$ Select the correct answer.

(Short Answer)

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Question 68

A rubber ball is dropped from a height of $8 \mathrm {~m}$ 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. Select the correct answer.

(Multiple Choice)

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Question 69

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

(Short Answer)

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Question 70

Find a power series representation for $f ( t ) = \ln ( 14 - t )$

(Multiple Choice)

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Question 71

Determine which series is convergent. Select the correct answer.

(Multiple Choice)

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Question 72

Determine whether the series is convergent or divergent. $\sum _ { n = 1 } ^ { \infty } \frac { \tan ^ { - 1 } n } { n \sqrt { n + 6 } }$

(Short Answer)

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Question 73

How many terms of the series do we need to add in order to find the sum to the indicated accuracy? Select the correct answer. $\sum _ { n = 1 } ^ { \infty } 2 \frac { ( - 1 ) ^ { n + 1 } } { n ^ { 2 } } ( \text { |error } \mid ) < 0.0798$

(Multiple Choice)

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Question 74

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 75

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.

(Multiple Choice)

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Question 76

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

(Short Answer)

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Question 77

Determine whether the geometric series converges or diverges. If it converges, find its sum. $\sum _ { n = 0 } ^ { \infty } 5 ^ { n } 6 ^ { - n + 1 }$

(Multiple Choice)

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Question 78

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

(Short Answer)

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Question 79

Use multiplication or division of power series to find the first three nonzero terms in the Maclaurin series for the function. $f ( x ) = 5 e ^ { - x ^ { 2 } } \cos 4 x$

(Multiple Choice)

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Question 80

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 $\sum _ { n = 0 } ^ { \infty } c _ { n } x ^ { 2 n }$ .

Determine whether the sequence convergent or divergent. $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 2 } - 6 n + 10 }$

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

Determine whether the sequence defined by $a _ { n } = \frac { \sin 2 n } { 9 n }$ converges or diverges. If it converges, find its limit.

Test the series for convergence or divergence. $\sum _ { m = 1 } ^ { \infty } \frac { ( - 6 ) ^ { m + 1 } } { 4 ^ { 8 m } }$

Which of the given series are absolutely convergent?

Determine whether the series is absolutely convergent, conditionally convergent, or divergent. Select the correct answer. $\sum _ { n = 1 } ^ { \infty } \left( \frac { 4 n ^ { 2 } + 3 } { 3 n ^ { 2 } + 4 } \right) ^ { n }$

Use series to evaluate the limit correct to three decimal places. $\lim _ { x \rightarrow 0 } \frac { 7 x - \tan ^ { - 1 } 7 x } { x ^ { 3 } }$ Select the correct answer.

A rubber ball is dropped from a height of $8 \mathrm {~m}$ 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. Select the correct answer.

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

Find a power series representation for $f ( t ) = \ln ( 14 - t )$

Determine which series is convergent. Select the correct answer.

Determine whether the series is convergent or divergent. $\sum _ { n = 1 } ^ { \infty } \frac { \tan ^ { - 1 } n } { n \sqrt { n + 6 } }$

How many terms of the series do we need to add in order to find the sum to the indicated accuracy? Select the correct answer. $\sum _ { n = 1 } ^ { \infty } 2 \frac { ( - 1 ) ^ { n + 1 } } { n ^ { 2 } } ( \text { |error } \mid ) < 0.0798$

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

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.

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

Determine whether the geometric series converges or diverges. If it converges, find its sum. $\sum _ { n = 0 } ^ { \infty } 5 ^ { n } 6 ^ { - n + 1 }$

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

Use multiplication or division of power series to find the first three nonzero terms in the Maclaurin series for the function. $f ( x ) = 5 e ^ { - x ^ { 2 } } \cos 4 x$

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Exam 11: Infinite Sequences and Series

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 .

Determine whether the sequence convergent or divergent. ∑n=1∞1n2−6n+10\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 2 } - 6 n + 10 }∑n=1∞​n2−6n+101​

Determine whether the given series converges or diverges. If it converges, find its sum. ∑n=1∞(1+5n)n\sum _ { n = 1 } ^ { \infty } \left( 1 + \frac { 5 } { n } \right) ^ { n }∑n=1∞​(1+n5​)n

Determine whether the sequence defined by an=sin⁡2n9na _ { n } = \frac { \sin 2 n } { 9 n }an​=9nsin2n​ converges or diverges. If it converges, find its limit.

Test the series for convergence or divergence. ∑m=1∞(−6)m+148m\sum _ { m = 1 } ^ { \infty } \frac { ( - 6 ) ^ { m + 1 } } { 4 ^ { 8 m } }∑m=1∞​48m(−6)m+1​

Which of the given series are absolutely convergent?

Determine whether the series is absolutely convergent, conditionally convergent, or divergent. Select the correct answer. ∑n=1∞(4n2+33n2+4)n\sum _ { n = 1 } ^ { \infty } \left( \frac { 4 n ^ { 2 } + 3 } { 3 n ^ { 2 } + 4 } \right) ^ { n }∑n=1∞​(3n2+44n2+3​)n

Use series to evaluate the limit correct to three decimal places. lim⁡x→07x−tan⁡−17xx3\lim _ { x \rightarrow 0 } \frac { 7 x - \tan ^ { - 1 } 7 x } { x ^ { 3 } }limx→0​x37x−tan−17x​ Select the correct answer.

A rubber ball is dropped from a height of 8 m8 \mathrm {~m}8 m onto a flat surface. Each time the ball hits the surface, it rebounds to 50%50 \%50% of its previous height. Find the total distance the ball travels. Select the correct answer.

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

Find a power series representation for f(t)=ln⁡(14−t)f ( t ) = \ln ( 14 - t )f(t)=ln(14−t)

Determine which series is convergent. Select the correct answer.

Determine whether the series is convergent or divergent. ∑n=1∞tan⁡−1nnn+6\sum _ { n = 1 } ^ { \infty } \frac { \tan ^ { - 1 } n } { n \sqrt { n + 6 } }∑n=1∞​nn+6​tan−1n​

Find the values of ppp 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​

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 geometric series converges or diverges. If it converges, find its sum. ∑n=0∞5n6−n+1\sum _ { n = 0 } ^ { \infty } 5 ^ { n } 6 ^ { - n + 1 }∑n=0∞​5n6−n+1

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​

Use multiplication or division of power series to find the first three nonzero terms in the Maclaurin series for the function. f(x)=5e−x2cos⁡4xf ( x ) = 5 e ^ { - x ^ { 2 } } \cos 4 xf(x)=5e−x2cos4x

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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

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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 $\sum _ { n = 0 } ^ { \infty } c _ { n } x ^ { 2 n }$ .

Determine whether the sequence convergent or divergent. $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 2 } - 6 n + 10 }$

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

Determine whether the sequence defined by $a _ { n } = \frac { \sin 2 n } { 9 n }$ converges or diverges. If it converges, find its limit.

Test the series for convergence or divergence. $\sum _ { m = 1 } ^ { \infty } \frac { ( - 6 ) ^ { m + 1 } } { 4 ^ { 8 m } }$

Determine whether the series is absolutely convergent, conditionally convergent, or divergent. Select the correct answer. $\sum _ { n = 1 } ^ { \infty } \left( \frac { 4 n ^ { 2 } + 3 } { 3 n ^ { 2 } + 4 } \right) ^ { n }$

Use series to evaluate the limit correct to three decimal places. $\lim _ { x \rightarrow 0 } \frac { 7 x - \tan ^ { - 1 } 7 x } { x ^ { 3 } }$ Select the correct answer.

A rubber ball is dropped from a height of $8 \mathrm {~m}$ 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. Select the correct answer.

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

Find a power series representation for $f ( t ) = \ln ( 14 - t )$

Determine whether the series is convergent or divergent. $\sum _ { n = 1 } ^ { \infty } \frac { \tan ^ { - 1 } n } { n \sqrt { n + 6 } }$

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

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

Determine whether the geometric series converges or diverges. If it converges, find its sum. $\sum _ { n = 0 } ^ { \infty } 5 ^ { n } 6 ^ { - n + 1 }$

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

Use multiplication or division of power series to find the first three nonzero terms in the Maclaurin series for the function. $f ( x ) = 5 e ^ { - x ^ { 2 } } \cos 4 x$